The principle of linear superposition holds for the integers, eg. 3 + 4 = 7. However, when we take the integers out to infinity we now obtain nonlinear behaviour. Infinity plus any amount is still infinity, so linearity has broken down.
It seems paradoxical that the system of integers possess linear behaviour, under addition, and then that linear behaviour fundamentally changes at infinity. How can linearity result in nonlinearity out of seemingly nowhere?
This seems fundamentally paradoxical to me.
A worrying aspect of this is that logical induction breaks down. By induction, we can predict linear superposition for the addition of integers for higher and higher values, and then at infinity the induction process collapses.
Is this not a philosophical worry, in general, for proofs by induction?
But if we define addition as a linear function: f(x+y)=f(x)+ff(y), where, e.g, f(x)=x, then
f(x+infinity)=f(x)+f(infinity)=x+infinity and linearity of those relations is not spoiled by infinity.:-)
Dear Derek,
The non-linearity stems from our minds, in which we have conjured up this idea that infinity is just A big number like any other big number. And the idea helps us to some extent and hence is persisted with, although philosophically it is a little unsettling.
The paradox can be avoided if we define infinity not as the largest possible number but as that which includes every number, i.e. it is the union of all numbers, and hence has nothing outside it to be added to but has everything inside it to be subtracted from. All the finite numbers, like the ones and twos, all come from that infinity, and are not available outside of it, for addition.
If something is outside it, then it can not be infinity, since that which is outside it bounds it and makes it finite. Even the subtraction of any finite quantity from it will make it finite (infinity - finite = finite).
Of course, the symmetry of addition and subtraction will be lost, but the concept of infinity will be free from non-linearity.
Regards,
Rajat
The paradox appears when one thinks of infinity as an entity (mostly a number). Infinity is not a number and "at infinity" is not a place that one can visit. These terms are merely notions for limit behaviour. For example, the statement
"x/(x+1) approaches 1 as x approaches infinity"
means the values of x/(x+1) can be made arbitrarily close to 1 by taking x sufficiently large. Note that x is still a number. In the case of integers, no inductive argument ever reaches infinity (you can always approach it though).
But if we define addition as a linear function: f(x+y)=f(x)+ff(y), where, e.g, f(x)=x, then
f(x+infinity)=f(x)+f(infinity)=x+infinity and linearity of those relations is not spoiled by infinity.:-)
Might this paradox mean infinity is a fallible human contrivance? And that any paradox is a demonstrable lack of true understanding for what is really going on? I'm no mathematician, but I do know that physical equations generating outputs of "infinities" and "zeros" for experimentally measured critical inputs usually means the application of that expression was wrong for the occasion. So,
Hi. First off, and you may have done this, study the difference between cardinal and ordinal numbers. What you say isn't quite true for ordinal numbers: omega plus 3 is not omega. Study surreal numbers too if you really want to go wild with thoughts of infinity. By the way, nobody's ever seen an "infinity" so you ought not to act as though it's just a given that it's "out there" somewhere.
I think the problem is that we cannot take the integers out to infinity and keep on operating as if infinity were a number… but it is not.
Infinity is a notion, and no matter how large an integer can be imagined, for ANY integer (irrespective of its value) the linearity still holds, and you can reiterate in the new largest integer just got from addition, and add on top again. Now you can repeat this "ad infinitum" and you will still not reach "infinity"…
In other words (but now far away from rigorous formulation), you could also say that infinity1 + a = infinity2, and linearity still holds.
Infinity is a notion easy for compact formulations and helping our mathematical structure, but it is not a regular quantity to operate with.
I would say induction rules remain safe.
Dear all,
have a look at www.theinfinitycomputer.com where a new approach to deal with infinity is discussed. It is shown that it is possible to deal with in such a way that infinity +1 is larger than infinity.
A lesson from physics:
Often occured singular expression in physics is 1/ (x-x_0) is handled formally mathematically as
lim_{\epsilon ->0 }1/( x-x_0+i \epsilon)= P( 1/ (x-x_0))-i \pi \delta(x-x_0 ),
where \epsilon is a small auxialiary quantity, \delta is the Dirac delta function, P is the principal value (for x\noneq x_0), i is the imaginary unit . Here, the singularity is avoided by taking the limit \epsilon->0. The delta function term yields a physical contribution when integrated, as is usually the case in physical problems. .
Nature does not like singularties and mathematics should respect this by taking limits instead of taking the infinity as a number. .
Hi, a more general definition of a quasi-particle could be that it should simply follow Newton's laws for particles. I.e. we need inertia, mass etc. E.g. wave packages behave like particles, independent on the character of the field. I am just thinking about internal-gravity waves which, when incoherent, behave possibly like a particle gas, collisions etc. etc. - makes life much easier than to play with wave equations. Bonsoir, Helmut
Derek, I'm not sure what do you mean by linearity of addition of integer numbers.
Look: if linearity of such an addition means that there exists a linear operator converting its two arguments (or one two-dimensional argument, if you prefer) into a number then
A = [1,1] is such an operator, acting on 2-tuples of integers.
Let's try:
A [\infty,1]^T = \infty, or in other words: \infty + 1 = \infty. No trace of non-linearity.
I believe that situation is even more simple than Ya. Sergeyev wrote, look at handbook by Kolmogorov and Fomin and find a part devoted to ordered sets you will see that there are so called transfinite numbers (like infinity + 1) which are base of extension of induction beyond infinity and has name of transfinite induction. I hope the book is exactly what you need to resolve the paradox...
In the attachment here entitled, "From linearity to nonlinearity and a maximum point, not infinity", I have discussed the behavior of actual experimental observations, (x, y) pairs, all expressed in whole numbers. Hope this is of relevance to the topic being discussed here.
There is the definition of infinity : all real numbers n fulfill the inequality n
The set Z of all integers is an infinite set containing all finite integers. On this set the operation + makes Z an abelian group. So addition, subtraction, etc., work as expected. Infinity is a mathematical concept. There is no element \infty in Z, so addition is not defined with \infty.
There is, however, the concept of ordinal numbers, and there is an arithmetic defined for ordinal numbers - ordinal arithmetic. This coincides on the natural numbers with ordinary arithmetic and there are different infinities. Interestingly, + is no longer commutative on the infinite ordinals. But that properties of operations get lost when the domain of definition is extended is not rare. So is, e.g., the multiplication on the quaternions not commutative, either.
As I found in Wikipedia "ordinals" are synonyms of "transfinite number" (see my comment), I agree with Hermann that operation + (mentioned in the original question) can be introduced only for ordered sets. The initial paradox of the discussion is totally resolved in "Elements of the Theory of Functions and Functional Analysis" by Kolmogorov and Fomin. -> In ordered sets concept of induction can be extended up to so called transfinite induction so "at infinity the induction process " does not collapse.
Free download, http://booksee.org/book/1177661
Eugene, that's true but for such sets the main question/paradox of the discussion is not applicable...
@Eugene, Sergey:
There is the Well-ordering theorem that states that every set can be well-ordered.
Derek, what Marek possibly has had in mind is something like A[x,y] = x/(1+x). If you additively increase x by y then, at high x, A remains almost unchanged - the trace of non-linearit you quoted.
Infinity is an enormously interesting concept for physicists and mathematicians. For physicists because our world is possibly infinite in time and space. Cosmological speculations about a heat death of closed systems are not applicable to a real (infinite) world.This focuses the forces of physicists on things which are more testable/measurable than the finite fate of the cosmos. Math gives us the precise notions and processing rules (i.e. what we call language) to express that understandably and uniquely. Exciting topic!
DSc Laxmanan, you wrote this "simple systems produce linearity, while "complex" systems are nonlinear." Well, depends what you are looking at. The simple heliocentric system uses a law of gravitation wherein Newton's force scales with the inverse of a distance - clearly a simple system (one Sun, one Earth) but with 1/r clearly non-liear. I'd insist to claim that "infinity" is an expression of physicists' (and cooks' and all practicioners') everyday practice to differentiate between "essential" and "negliglible". Math has put it in the right and exact language, obviously a hard job which took couple of centuries.
Eugene, great argument, the night sky is empty? Well, we see only the finite stuff. What is actually thrilling me is the problem of infinity and closedness. Can an infinite system be closed with respect to energy+mass and charge etc.? Infinity, conservation and symmetry. I am afraid I need a seminar with Mdm. Noether and not with students of her students ...
Helmut,
Mdm. Noether has written all what she whanted. Seminar with her is no problems. Also with Hilbert. I forgot the name of that person, after whom is night sky paradox. He was the friend of Gaus and Bessel. The sense of this paradox is not in what we see, but in what happens. If You multiply the number of stars R^3 by power of light from them 1/R^2 You receive R, i.e. infinity for infinite (R) universe.
Regards,
Eugene.
Eugene, It's Obers, see http://en.wikipedia.org/wiki/Olbers%27_paradox
My concern is more thermodynamics of the universe wherein we OBSERVE certain conservation laws with accuracy, and with Mdm. Noether, deduce corresponding symmetries, which all work only IF we talk about finite volumes in space-time. Here we possibly have an interesting and crucial trap for the notion of infinity in physics and math, which I think was a great question posed by Derek Abbott. I am not sure whether I can contribute here as I have some expertise only in another field. However, the more fundamental a question, the simpler and less complex it is and much depends on other factors ...
When I was reading Kripke’s famous article “Outline of a theory of truth”
http://philo.ruc.edu.cn/logic/reading/Kripke_%20Theory%20of%20Truth.pdf
years ago, I thought that there must be a discontinuous way in progressing upwards because Kripke says that the fixed points come in after a certain point: the transfinite level. – Why not before? I understood that it was about numbers that do not have a predecessor. Is it the only difference? Is nonlinearity relevant here?
To refer the problem shortly: the traditional solution to the Liar Paradox was to repeatedly step up one sentence in the hierarchy of meta languages. - Kripke’s solution is to stay in the same language and to provide a fixed point i.e. to have a sentence that is able to claim its own truth (= fixed point). Therefore he assumes that the set of all sentences of language L is exhausted at one point. His theory makes it possible that a language contains its own truth predicate. Nevertheless Kripke uses an object language and a meta language - but not an infinite hierarchy.
To make my question clearer: that the fixed points come in (are available at a certain point of the development and not before) - is it dependent on linguistic or on mathematical structures? – Thanks for any suggestion.
We must better set foundational Basis before speaking about those things. In fact, there are a lot of Systems of computation where " infinite elements" can be added, and more or less linearity is still working. Examples:
- Cardinal computation. Here the sum has the sense "cardinality of the union of infinite sets". $\aleph_0 + 2^{\aleph_0} = 2^{\aleph_0}$. Very nonlinear.
- Ordinal computation. Mean to concatenate ordinals (well ordered set) such that the Union (+ order type) is still an ordinal. In general linear: one has $\omega + 2^\omega = 3\omega$
- Computation in non-archimedean ordered fields - as for example the field of non-Standard real numbers, but also other fields, like Q(X) ordered so, that X > N. Here all is linear, also for the infinitely small elements.
- A particular case of the last case: Conway's field of all numbers (of all sets!) It contains equally all ordinals and infinitely small elements, and is a real closed field, like the reals.
- AND SO ON.....
So, instead of putting philosophical questions, we better choose the System which best satisfies our needs, and we keep working
Is night sky paradox an opposite version of Zeno’s “One and Many Paradox”?
Geng,
night sky is about what we can observe (Nature), but Zeno's paradox is purely logical, about what can exist.
Regards,
Eugene.
P.S. For Zeno infinity can not exist a priori (absurd in Greek's language).
Eugene, both “night sky paradox” and Zeno’s “One and Many Paradox” are about logical and quantitative.
Geng,
"night sky" is about what we see, but "one and many" about what we think. It is different things.
Regards,
Eugene.
"Infinity plus any amount is still infinity, so linearity has broken down" It is a quotation from Derek. But if we assume that (inf + 7= inf + 7) and, for example: (inf + 7) is not equal to (inf + 8) and so on, then , I think - inearity - and many other number properties can be preserved although I do not know whether it has some deeper meaning:-(
For more than 2500 years since Zeno’s time, people working very hard on the surface, but never touched the defects disclosed by Zeno’s Paradoxes-------in front of Zeno’s Paradoxes, our science and wisdom have been frozen and we still live in Zeno’s time.
Hi Geng, I dont agree that "we still live in Zeno’s time" because we see more than Zeno and its contemporaries. For me, the error in the paradox of Achilles unable overtakes a turtle lies in the fact that the Zeno does not take into account the different velocities of the two runners. He could not do this because in those days there was not quantified definition of velocity:-)
Jurek
Dear Jerzy,
I am very sorry to say that so far no one really solve the defects disclosed 2500 years ago by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race”.
The defects disclosed by this paradox family are the logical contradictions between so called “potential infinity” and “actual infinity” theoretically and practically, but not only how to calculate when or how Achilles catch up with the Turtle; the simplest similar resolution is something like 1+1/2+1/4+1/8+…=2, or a real race between Mr. Smith and an alive strong Turtle.
And, the typical modern version of this Zeno’s Paradox is the Paradox of Harmonious Series.
Sincerely yours,
OUYANG Geng
Dear Quyang!
Thank you for your detailed and interesting answer, but I recall from my memory the lectures of philosophers who argued that this paradox is resolved. Maybe, they had in mind different logics, for example such which took into account the time element? One thing is certain - an infinity, is very interesting idea:-)
Sincerely yours
Jerzy H
Jerzy, Geng,
resolution or not of this paradox is the matter of beliefe. Exists infinity or not?
Regards,
Eugene.
Can we use the “brackets-placing rule"(１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．) applied in the divergent proof of Harmonious Series (1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．) to produce infinite numbers bigger than 1/2 or 1 or 100 or 100000 or 10000000000…?
Brackets placing rule is valid only for absolutely converging series, Geng.
Regards.
Eugene.
“Achilles and Tortoise” Zeno’s Paradox has disclosed the "real infinite--potential infinite" fundamental defects in present traditional infinite related science theory system. In this theory system, the critical defects are unsolvable because we are not allowed to forget either "real infinite" or "potential infinite" in such paradox (exactly same situation happen in Harmonious Series Paradox).
That is why this paradox family has been troubling us human for more than 2500 years.
No it is not broken . Infinity has strange properties, it is even and odd....
Article New Philosophical View Merging Infinity, Imaginary number an...
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