How can we introduce the idea of infinity to students? Its properties, relationship with zero etc.?
In some sense we can say that:
Infinity is a number beyond our counting numbers, having following properties:
x÷0=∞,x≠0
0+∞=∞
0-∞=-∞
∞-0=∞
0×∞=undefined(Think?)
∞÷0=∞
0÷∞=0
∞+∞=∞
∞-∞=undefined (Think?)
∞×∞=∞
∞÷∞=undefined(Think?)
A nice topic is the Hilbert hotel with countably many rooms. If it is full and someone comes late, then this person can still sleep if everyone shifts to the next room. If on the opposite side of the street there is a similar hotel also filled up and it burns down with everyone saved, then these infinitely many people can still sleep after the correct rearrangement. However, the powerset of the natural numbers cannot fit into the Hilbert hotel.
Infinity is to some degree a byproduct of other work in say number theory when the first irrational number was noticed to exist and any transcendental numbers which have nonrecurring decimal places that appear to go on forever but not repeating. Something similar occured when divergent sequences and series were first noticed and also when the first ideas of convergence could be constructed. In this sense infinity is something that does not stop but goes on unendingly.
How about, leveraging the studnets familiarity with the concept of zero, thus:
1. Begin with a discussion of the concept of zero
2. Discuss its potential (i.e. in some cases) practical intangibility and maybe get some laughs, for example:
a) You could say, "imaging I have one apple in one hand and minus one apple in the other hand. I add them together and I now have no apples ... AAAARRRRHHH".
b) Here I have a zero. Here I have another zero. I add them together I still have zero. Nothing in reality will prepare you for doubling something to get no more than what you started with. However, there is another concept that has the same property, but it lies on the other extreme to zero and we are now going to look at how we can conceptualise it
3. Consider the idea of reciprocals: y = 1 / |x|. Let's look at some of the properties of this function:
a) If the absolute value of x is greater than one, then as x increases, y decreases.
b) Now, if the absolute value of x is less than one, then y increases.
If your students have already been introduced to differentiation, then they will know about the idea the 'limit as x approaches...' as in, for example a discussion of tangents. If not, it may be a good idea to do that first.
Hence,
4. Infinity is the limit of y (in the above equation) as x approaches zero.
5. Discuss some of the properties it shares with zero. For example,
a) you can double it and it won't double!
b) You can add it to any number but it will not change.
Now for some fun:
6. Introduce Zeno's paradox of motion:
Sum of 1 to infinity of 1/(x squared) = 1, otherwise motion would be impossible!
Have fun!
As we know, there are two types of infinity! Well, okay, infinitely many types of infinity, but what I mean is the infinity that we encounter in a Calculus course is, philosophically speaking, different from the type of infinity that we encounter in a Basic Set Theory course. The infinity in the former context can be formally avoided altogether, and introduced as a mere symbol. The precise definitions of limit at infinity, etc. do not mention "infinity" at all. For example, "a_n approaches 2013 as n goes to infinity" can be rephrased without mentioning infinity. In that context, infinity simply allows us a convenient "figure of speech", so to speak. In the context of Set Theory, however, we are dealing with what is known as "actual infinities" (as opposed to "potential"). The Hilbert Hotel example mentioned above is a good an entertaining introduction to the mysteries of that sort of infinity. I believe historically it was Galileo Galilei who first noticed that there are as many positive integers as there are perfect squares!
Dear Amir,
The points I wish to study are:
1. We do not want to avoid infinity, but to study it.
2. How many types of infinity are? Countable, cardinality of interval [0,1] and ...
3. Why infinity has these wondeful properties some of which are mentioned above.
4. Most important of these is how to give this idea to undergraduate students of set theory/logic/calculus
Ismat.
Let me recommend you this paper:
Infinity: The Twilight Zone of Mathematics.
By WILLIAM P . LOVE (1989)
Mathematics Teacher, pp. 284-292
Good luck!
The copyright's owner of Love's paper is, if I remember accurately, the National Council of Teachers of Mathematics (Reston, Va, USA).
To introduce the concept of infinity to a starter, I would suggest the following way. We cannot complete writing the sequence of natural numbers 1, 2, 3, ..., because it is infinite. This infinity is the smallest of all infinities in the sense that the number of terms in this sequence is 'countably' infinite. On the other hand, between two numbers, say 2 and 3, we can see that there is always an average, in our example 2.5. Now between 2 and 2.5, there is another average, 2.25. We can go on finding such averages. This process can not be completed because it will continue to eternity. In other words, this sequence of the averages is already an infinite one. Accordingly, the number of points in the entire interval from 2 to 3 would be infinite, but not countable. This is technically an infinity of the largest kind. Students would then find it easy to grasp the idea of infinity. Now in a line segment there will be an uncountably infinite number of points each of which is of length zero. So in this case, zero multiplied by infinity would be the length of the line segment which is finite. Therefore zero multiplied by infinity is not necessarily an infinity. I believe, this is how the concept of infinity can be introduced to young learners.
This small book may give you some ideas:
http://www.introducingbooks.com/book/view/infinity-a-graphic-guide
The question was how to introduce the idea of infinity. Obviously, the audience are young learners. Therefore the system of introduction must be plain and simple as far as possible. The students may be told to draw the graph of y = 1/x, the rectangular hyperbola. After plotting a few points, they would realize that as x approaches zero, for positive x, y approaches infinity, and that for negative x, y approaches minus infinity.
I usually introduce it by reminding people that young children can't count very far, but even if a child can't count past 10, they can still tell if a pile of 20 letters is the "same size" as a pile of 20 envelopes, because all they have to do is put one letter in each envelope and check that nothing's left over. I then suggest that "counting to infinity" is no different - even as adults, we don't really know how to count very far, and we call things we can't count "infinite" - but this doesn't stop us knowing when two different infinite collections are the same size. This leads easily to the idea of 1-1 correspondences, and then I do the standard example of the reals not being countable. Seems to work!
Use the pigeonhole principle: if n pigeons are put in m pigeonholes, the unique possibility to have at most one pigeon in one pigeonhole is when m is greater or equal than n, but this is not the case for infinite sets. Notice that when n equals m this gives rise to a one-to-one correspondence. But in infinite sets, this is not necessary. You can map naturals in even naturals: n -- 2 n having an injective but not surjective correspondence.
Mathematically strict approach to the definition of infinity you can find, for example, in the book:
Kenneth KUNEN. "SET THEORY": AMSTERDAM - LONDON - NEW YORK -TOKYO (1992)
see page 19 (the definition of the set of natural numbers and axiom of existence of this set) and page 28 (definition 10.8 of infinite sets).
Ismat,
For example you have one point (.) there is infinite position to view them, you can watch them to near, far, behind and from any location because this is exit there but you can see them from the different locations. if your position is fixed and move the point. then point is exist. and point have infinite location.
other example for example number cannot exist in the world. then what is infinity, there is only 3 things absence, presence and many , but how can you define many. a things you are unable to count them, calculate them. is infinite.
The following book may be a good reference for that:
Roads to infinity, By John Stillwell
Since infinity is such a vast and intimidating topic, we prefer to begin by closely looking at the familiar ideas of numbers and counting. We want to count infinite collections, but we don't know how to count that high. Thus we seek different ways to count collections of ordinary objects in the hope that one method might work for infinite collections as well.
The fundamental idea in the study of infinity is that two collections have the same size if there is one-to-one correspondence between the members of one collection and the member of the other collection.
Qefsere's answer shifts the question to infinite sets. Cardinals of infinity sets are infinite number: alef-zero, c, ...An infinite set has the remarkable property that she remembers to us; for instance, there is a bijection (one to one correspondencee) between even numbers and natural numbers although the set of even numbers is properly included in the set of natural numbers. I respect every point of view, but I prefer the amazing simplicity of Ismat question, related to the idea of infinity; in set heory "there exists an infinite ser" is an axiom!
Along the years, the concept of the infinite was target of a fervent discussion. Since Ancient Greece, Men have tried to come up with a definition for that concept. Demócrito, Zenão, Aristóteles, Arquimedes, Galilei, Bolzano, Dedekind, Cantor, Weierstrass, Poincaré, Hilbert, Borel, Russel, Robinson... these are just a few of the mathematicians that have dedicated themselves to this topic. The history of the infinite as a mathematical entity is long and controversial. The infinite was always associated to several paradoxes situations. Slowly, some of those paradoxes were eliminated, but nevertheless it remains as a very subjective and complex subject.
We can introduce the idea through history.
The infinite was always present in Maurits Escher’s work (1898-1972), but around 1956 his work showed more clearly the representation of the infinite in a bi-dimensional background. We can divide his work on this subject according with three categories: cycles, filling in surfaces and limits. In each one, there are pieces that even with different characteristics can still be considered important and well known. Deeply connected to the concepts of cycles and filling in surfaces, we find the notion of Potential infinite, represented by an endless process. There is a systematic idea that suggests an unlimited process. Limits are associated with an isomorphic reduction of the images. In the pieces where he performed a reduction from the inside to the outside (bigger shapes go into the centre and the infinite reduction is in the margin with a circular shape), he produced a series called Circular Limits, made of four pieces. Amongst those we can find Circular Limit III (1959) as being Escher’s best approach to the infinite, reaching the Actual infinite.
We can introduce the idea through art.
Potential infinity es easier to explain than actual infinity. Potential infinity is related to any "repeating" process that never ends in your head, although it ends in the physical world. Let me give two examples.
(1) You may mix two colours according a mixing pattern. Your process do actually finish in the practical realm, but you may continue the process "indefinitely" in your head. Incidentally, remark that the observated mixed color will never change after 3 or 4 mixings, but in your head the new mixed color must be different, as the proportions change! (M. Coriat, P.S. Martinez, and J. Baena, 1993. Numbers and Colours. Int. J. Math. Educ. Sci. Technol., vol. 24, Number 4, pp. 501-510)
(2) You may half repeteadly any given quantity. For instance: number 1. You obtain, successively, 0,5; 0,25; 0.125;... Once again you will attain the frontier of your pocket calculator or your paper's end. However, you feel and you are sure that you can continue in your head.
We know all that from Aristotle's Physics. Modern, "constructivist " mathematicians accepted this infinite but rejected the other (actual infinite).
Remark: Constructive mathematics must not be related anyway to constructive edicational issues.
If you are able to apprehend entirely an infinite collection, your are thinking on actual infinity.
Two mirrors - one looks to another.
In them there is no reflection, but infinity.
What does this mean? What is it?
How to explain? - Mystery of the mind.
Perhaps there lies the road to infinity?
Can’t see through the glass end of the road.
Unfortunately, in every science concept of infinity is different.
What kind of science is it?
Pythagoras in the ancient Greece did believe that numbers constitute the true nature of the things. However, the big question is: whether the things which "exist" mathematically also exist physically...?
Infinity?
According to legend, St.. Augustus, while walking along the shore, and meditating on the infinite, he saw a child who was trying to empty the ocean with the help of one shell ...
Augustus De Morgan (1806-1871)
Legend of ancient India : raven 1 time in 1,000 years preen about diamond rock until it crumble. This time is negligible compared with infinity.
.
There are numbers out there that are so enormously, impossibly vast that to even write them down would require the entire universe. But here's the really crazy thing...some of these incomprehensibly huge numbers are crucial for understanding the world. you can look at the following link..it might help a bit
http://gizmodo.com/5809689/a-brief-introduction-to-infinity
There's a wonderful book called "Infinity, Beyond the Beyond the Beyond," by Lillian R. Lieber, illustrated By Hugh G. Lieber. It was first published in 1953, but it's back in print. It's available from a number of sources online, and I recommend it.
One can explain about infinity like this " If one can stand on the middle of the railway track, he may observe that the track coincide at some point. But originally it does not happen. Where the point he observed may be the infinity point".
I agree with Joseph. Often I read in the popular cosmology literature that the universe may be "finite", really meaning that it is bounded in the mathematical sense. I guess this imprecision in language can be confusing.
NATURAL NUMBERS AND the idea od INFINITY. Several years ago I read this remark. (I apologize as I forgot the source.) Any natural number is farther to "the other extreme" than it is to 0.
"Graphically":
0-1-2-3--------------------------------------------------------N---------------------------------->
Natural Intervals approach:
To the interval [o,N[ contains ALWAYS less numbers than the interval [N, ...[
Infinity is just the subject matter of mathematics. There is no infinity in reality, so mathematics is the science of unreality. Mathematical models are by nature approximative.
Hi Sreenu Sree,
It is very nice:
"Most things we know have an end, but infinity does not have an end"
What is the "end" ? Quantity or a qualitative change from one state to another
( In this universe in due time People and flower turn to dust , The Rubaiyat
By Omar Khayyam).
See http://lesswrong.com/lw/g74/can_infinite_quantities_exist_a_philosophical/,
it is very interesting.
Dear all,
see this book:
Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity
We should discriminate between "potential infinity" and "actual infinity". From ancient Greece until almost Gauss, infinity was identical with potential infinity. Let us denote,
potential infinity as, 1,2,3,...,n, where n is a very large natural number.
Actual infinityas {0,1,2,...}
Actual infinity is Cantor's central notion.
For potenetial infinity, we have Internal Set Theory, where a nonstandard number is a very large number, e.g. 2^100. In this way infinity is identical with incomprehesible large number. In mathematics we also have such nonstandard unlimited numbers, e.g. in Poincare model of non Euclidean Geometry. There a circle of finite radious can be seen to incorporate a kind of infinity. Let M be the disc. Then an inhabitant of M would feel that his universe is innite in extent on the simple grounds that he never reaches a boundary after taking any nite number N of steps, no matter how large N may be chosen.
Thus there are examples, where a finite set can be seem us unlimited, i.e a non Cantorian infinity.
Although intangible assets with finite rather difficult to explain the concept of infinity, the correct teaching technique can be used from simple to complex. In the minds of all people with a sense of infinity is surely and everyone would agree the concept is difficult to create. For the teaching of this concept is very important to choose a suitable age. With this in mind primarily about how to approach students to try to understand the concept of infinity is required. Then you will not confound the mind with simple examples can gain an insight into this concept. Especially in mathematics in my opinion one can begin it by using of convergence of infinite sequences and series or natural numbers. In primarily age this concepts will be given little by little and later it will be considered depthly. Therefore this concept can be created in a suitable way.
Dear Charles, you are right. Actual infinity cannot be found in nature. However imagine that the real world is formed in many "levels of reality". In each level corresponds a cardinal. The counting is unlimited, and suddently you are transfer counting in another level of reality, say where the cardinal is c. In this way we can use a mixture of potenetial infinity together with levels of reality in order to simulate the cardinals of Cantor. In any case potenetial infinity seems to have as a limit an actual infinity. For all these see Vopenka's paper:
There is the infinite and infinity. Infinity is mathematical. An infinitesimal, the inverse of infinity, is mathematical. Neither should be confused with the properties of the infinite.
A point on a Normal distribution has a probability of zero even when it is at the most probable location of the distribution. This mathematical result derives from an infinity of points (to achieve the infinitesimal) at that location.
An electron of an atom can interact with a photon of sufficient energy to ionize the atom. The electron has been removed to mathematical infinity, a distance that can be very small. For some applications that distance is infinitesimal.
Infinity is a mathematical concept having nothing to do with the infinite. Infinity is a number whose value if changed by 1(one) will have no appreciable effect on the outcome of the calculation. The summation of an infinite series can reach infinity at n = 20 or less.
The biggest problem with infinity is when it is invoked in the calculation or assumptions. Call on infinity at the wrong time and you have a paradox.
Achilles and the tortoise invoked infinity in the assumptions. Achilles could easily out run the tortoise, but was tripped up by the assumptions.
What to teach a student?
Infinity is a mathematical concept having nothing to do with the infinite. Infinity is a number whose value if changed by 1(one) will have no appreciable effect on the outcome of the calculation. Use with caution.
I think of infinity as a process, rather than a place, because no such place exists.
The process, of course, is to keep adding, multiplying, subtracting or dividing to your last answer and repeat the process without limit.
By the common N+1 property: Take a number N, as large as you want, add +1 and you have a greater number (For the simple cardinality).
Infinity is a relative measure, for example, far away from the earth is infinity for us and it counts thousands of km and light years to go far from the influence of the earth, but the infinity of an electron around the nucleus is just very small distance of nanometers and go away from the influence of the atom.
I found very interesting article:
A. W. More A Brief history of Infinity
Scientific American, april, 1995, p. 112
Dear Mehdi!
Look: Symbol of Infinity, http://en.wikipedia.org/wiki/Infinity_symbol
It is very interesting.
From http://en.wikipedia.org/wiki/Infinity_symbol;
The shape of a sideways figure eight has a long pedigree; for instance, it appears in the cross of Saint Boniface, wrapped around the bars of a Latin cross. However, John Wallis is credited with introducing the infinity symbol with its mathematical meaning in 1655, in his De sectionibus conicis. Wallis did not explain his choice of this symbol, but it has been conjectured to be a variant form of a Roman numeral for 1,000 (originally CIƆ, also CƆ), which was sometimes used to mean "many", or of the Greek letter ω (omega), the last letter in the Greek alphabet.
Symbol used by Euler to denote infinity
Leonhard Euler used an open variant of the symbol in order to denote "absolutus infinitus". Euler freely performed various operations on infinity, such as taking its logarithm. This symbol is not used anymore, and does not exist in Unicode.
To see Jon Wallis go to;
http://en.wikipedia.org/wiki/Infinity_symbol#mediaviewer/File:John_Wallis_by_Sir_Godfrey_Kneller,_Bt.jpg
I've always found infinite amusement in the idea of "adding a point at infinity" to the infinitely extended real line and getting something conceivable, like a circle. The classic stereographic projection makes the infinite plane smaller by enlarging it into a sphere. Sort of a mathematical oxymoron...
Dear Paul,
You should maybe mention, Poincare disc, which if see it from "outside" its radius is finite, but from inside, and for the entities embodied in it is infinity!
The same phenomenon we encounter in Nelson's IST and Vopenka's AST as well.
“How can we introduce the idea of infinity to students?”
When we want to introduce the idea of infinity to students, we should be sure first “potential infinity "or “actual infinity ", than troubles begin for ourselves at once because of the confused definitions of “potential infinitude” and “actual infinitude” (long-drawn-out and ceaseless “potential infinity--actual infinity” debate since antiquity proves this conclusion)….
“How can we introduce the idea of infinity to students? Its properties, relationship with zero etc?” is a very good but tough question because it touches the basic stones of 2500-year old present traditional infinity system: (1) infinity conception, (2) infinity related number forms as well as (3) infinity related number treating theory and techniques (limit theory). But it is our duty to introduce the idea of infinity to students! I think on one hand we introduce the present traditional idea and the successful operations in applied mathematics, accept the successful facts; on the other hand, we may encourage students to ask questions….
we may introduce this way.
There are three parts for infinitude concept:
1, infinitude is a kind of existing laws (natures);
2, the existing law (natures) of infinitude must be expressed by infinitude carries in human science such as infinitude related number forms in mathematics;
3, the infinitude related number forms in mathematics are treated by limit theory (applied infinitude, many "x---->0" infinitude related number forms treated by limit theory become zeroes).
Can anyone express self-justificationly what infinity, potential infinity and actual infinity are? If not, “How can we introduce the idea of infinity to students? Its properties, relationship with zero etc?” So, our students will never know what infinity is and our students’ students will never know what infinity is…., this is a tragedy!
I sincerely hope that RG can help to solve this problem.
I agree with Charles. My view of mathematics is as a distillation of our intuitions about the so-called real world. We as humans are limited by our perceptual equipment, but our cognitive minds push the envelope. (E.g., our eyes can perceive only a certain range of color, but we infer an invisible spectrum and build instruments to detect it.) Our view of "infinity," "straight line", "computable process," even logic itself, are all based on naïve intuition. What matters is the universal agreement; even the "agreement to disagree," as between classical and intuitionistic logicians. Introducing the concept(s) of infinity to students should always have this in mind.
I agree with toy Paul. I could add that what is our intuition about "infinity" is large incomprehensible, sets". This can give a kind of non-cantorial infinity, found in Nelson Internal Set Theory and Alternative Set Theory of Vopenka. The generalisation of this "physical infinity" gives the cantorian infinity.
Step by step, we human have been trying hard all the way to cognize so many things in universe, “infinity” is one of the things for us human to cognize. I don’t know now much we could do, but I am deeply convinced that this is something there waiting for us to do and, we should do.
My researches told me that Infinitude is really a very challenging and tough thing which many people would not touch or have to avoid. However, many facts have proved that the concept of objective rooted Infinitude exists in our science; it is right there and the infinitude things are there for us to face, cognize and treat. Because of the fundamental defects in present traditional Infinitude theory system, we human really meet many difficulties when working in the field relating to Infinitude.
To introduce the “idea of infinity” to students is really very easy as so many colleagues above have done. But it is difficult to come to real life with the” idea of infinity” when facing and treating the “number forms relating to infinity", especially their relationship with zero; for example, why sometimes those X--->0 are allowed to be zero by limit theory in many actual calculations but sometimes not, such as in newly discovered Paradox of Harmonious Series as well as in ancient Zeno’s Paradox’s “Race of Achilles and Tortoise” .
Dear Charles,
Your post on Zeno’s Paradox’s “Race of Achilles and Tortoise” impressed me deeply. If that Zeno’s Paradox is still unsolved, there must be many modern versions of Zeno’s Paradox challenging us human.
What do you think？
As we know, we now really have 4 confusing “infinitude” related things in our science:
(1) potential infinity as something unknown, as the pre-Aristotlean Greek believed,
(2) actual infinity as a property of sets, as Cantor thought on it,
(3) infinitesimal with the number form of X--->0,
(4) infinity as big number, as the pre-K12 child think on it.
We have a big trouble:
for the first 2, we have long-drawn-out and ceaseless “potential infinity--actual infinity” debates since antiquity; while for the last 2, people use them as “numbers relating to infinitude” in many kinds of practical calculations but they are “non-number number of variables（"One thing is important"-------theoretically they should not be numbers but "another thing is also important"------- practically they should be numbers）”.
We human have been “opening one eye and closing another eye” ever since.
What can we do?
Geng Ouyang
As is acknowledged by most modern scientists that there are two kinds of “infinite” in our science: “potential infinite” and “actual infinite”. So, it means there are two kinds of “infinite things” in our science. Very possibly, if our students ask us how to justify the exact “infinite things” in our science “potential infinitude” or “actual infinitude” (such as the “infinite things” in Hilbert's Hotel, infinitesimals in calculus and many infinite serials), what can we do?
How can we introduce the nature of “infinite” to our students, can we have different degree of “infinite”------ more “infinite”, more and more “infinite”,…; less “infinite”, less and less “infinite”, … ;…?
I have been seeking the answer for years, can anyone help?
Talking about Cantor's view of infinity, I am a little sorry to say that he cared little for infinitesimal. Does infinitesimal belong to infinite and is infinitesimal anything to do with cardinal numbers or ordinal numbers?
So, that is why I call Cantor's view of infinity “half infinite”; I know many people nowadays will not agree with me, but what I have said is true.
Theoretically and practically, no rigorous mathematics treatments of infinitesimal work to solve the defects discovered 2500 years ago by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race”.
This is why I say that we are still living in Zeno’s time 2500 years ago'.
Mathematics or reality?
Dear Charles,
Thank you!
But according to my studies, non-standard analysis can do nothing to solve the defects discovered 2500 years ago by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race”.
“By saying that mathematics need only approximate reality” was the way people have been doing since Zeno’s time.
Regards!
Geng
Dear Charles,
The zeno paradoxes essentially refer to abstract mathematical entities rather than entities of nature. The use of Achilles and tortoise is just for decoration purposes! The actual things are just "distances", mathematical objects.
Dear Charles,
You are right; limit theory is a kind of special “quantity cognizing treating theories and techniques for the infinite things (especially for infinite related number forms)”. But the suspected defects disclosed by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race” tell us that limit theory is unable to offer the calculus a rigorous foundation:
"There is no limit to a limit" as the tortoise quipped to Achilles after the race, "since for any delta we can always find an epsilon such that delta is not small enough".
"But then", said Achilles, "all I will have to do is choose a smaller delta, and the result will be closer than your new epsilon". "In that case, answered the tortoise I will give you another epsilon, such that it is not". And so it was that even though Achilles and the tortoise managed to complete the race, they never finished the argument.
Regards!
Geng
For Geng:
In the article of Stanford Encyclopedia:Zeno's Paradoxes we find:
"Non-standard analysis: Finally, we have seen how to tackle the paradoxes using the resources of mathematics as developed in the Nineteenth century. For a long time it was considered one the great virtues of this system that it finally showed how to do without infinitesimal quantities, smaller than any finite number but larger than zero. (Newton's calculus for instance effectively made use of such numbers, treating them sometimes as zero and sometimes as finite; the problem with such an approach is that how to treat the numbers is a matter of intuition not rigor.) However, in the Twentieth century Robinson showed how to introduce infinitesimal numbers into mathematics: this is the system of ‘non-standard analysis’ (the familiar system of real numbers, given a rigorous foundation by Dedekind, is by contrast just ‘analysis’). And it has been shown by McLaughlin (1992, 1994) that Zeno's paradoxes can also be resolved in nonstandard analysis; they are no more argument against non-standard analysis than the standard mathematics we have assumed here. It should be emphasized however that—contrary to McLaughlin's suggestions—there is no need for non-standard analysis to solve the paradoxes: either system is equally successful. (The construction of non-standard analysis does however raise a further question about the applicability of analysis to physical space and time: it seems plausible that all physical theories can be formulated in either terms, and so as far as our experience extends both seem equally confirmed. But they cannot both be true of space and time: either space has infinitesimal parts or it doesn't.)"
It looks that McLaughlin (1992, 1994) solves the paradox using infinitesimals.
Since I am sure that Dear Geng did not read carefully the following paper or you did not have access I just attached it!
Dear Charles,
OK, nature does not contain "continuum"! But what is continuum? There is of course the classical absolute answer, but there is a "phenomenological" one! Let x~y means that I cannot discriminate with my eyes x from y. Then the "continuum" is a phenomenon. The surface of a table "seems" continuous! The same is true for my computer monitor: Using mathematica I "see" a continuous curve. Then there is a fundamental question: what is the difference of what I see in my computer monitor and what I "see" in my fantasy-mind monitor?
These views are rather non Cantorian, and are related to Alternative Set theory (AST) and Internal Set theory (IST). The article I attached in my previous Post, uses IST.
Dear Charles,
What you are saying essentially amounts to Cartesian duality. The outside world lies over there (external) and my mind is "internal". Lately they use the Theory of Affordances of J. J. Gibson, to attached to every living subject its niche, the space that afford to do things appropriate fot the subject. Taking also the Darwin theory of evolution, each intelligent subject have been developed as a unit with its niche. The intelligence depend on the niche or the affordances. Mathematics is a part of this unity. For more information see:
Danielle Macbeth, Realising Reason: A Narrative of Truth and Knowing. Oxford 2014.
I also attached Vopenka;s paper, where the natural infinity is treated.
Great thread, Charles and Costas! I think Charles' answer regarding physics and mind is actually a nondual view. We observe the phenomenal world--such as we conceive of it--but the question of its independent existence is not really relevant to our endeavors. Buddhist mystics have been writing about this for centuries; as, no doubt, have others. The conceit behind the movie "The Matrix" posits a "phenomenal world" which is a machine construct: the whole of "reality" is just very elaborate code. So I seem to agree with the idea that both mathematics and physics are expressions of Mind looking at itself and what it thinks is physical reality. (These ideas really need a poet to express them more adequately.)
Dear Mr. Costas Drossos and Charles，
Thank you so much, you are just like knowledgeable brothers, always very kind and helpful.
To Costas, I have really read the papers you recommended, so many people have offered ideas and methods to “analysis, solve” Zeno’s Paradox (it is easy) since antiquity, but the problem is in front of the practical case they are unable to do anything. When we ask: how and when dose Achilles catch up with the Turtle? No one in the world so far can offer rigorous answer theoretically and practically. This is the point.
To Charles and Costas Drossos，
In present traditional “potential infinite--actual infinite” based infinite theory system, in front of non-standard analysis, standard analysis with limit theory is not rigorous and, in front of Zeno’s Paradox of “Achilles--Turtle Race”, standard analysis and non-standard analysis are impossible to be rigorous.
Sincerely yours,
Geng
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