Euler's paradoxical series is a nice example.

1, -1/2, +1/3, -1/4, ...

Add the first n terms of the series, you get some number depending on n, and as n gets larger it gets closer and closer to log(2).

If you add only the first n positive terms you get a sum which converges to +infinity as n -> infinity.

If you add only the first n negative terms you get a sum which converges to -infinity as n -> infinity.

By adding more and more negative terms, and more and more positive terms, but taking different numbers of positive and negative terms, you can arrange that the total converges to anything you like between - infinity and + infinity, by taking different balances between numbers of positive and negative terms.

For instance, with the ordering

1, 1/3, -1/2, 1/4, -1/5 ...

you always have either two or three more positive terms than negative terms.

I think that once you realize that the sum of the positive terms is infinite and the sum of negative terms is negative infinite, the paradox kind of disappears. There is clearly a huge reservoir of different possibilities, determined by choosing different balances between negative and positive terms.

There is of course a theorem which says that the limit of the series is the same under any reordering of the terms if and only if at least one of the two series (sum of positive terms; sum of negative terms) is finite.

If both positive and negative series diverge but the individual terms themselves converge to zero, then by suitable reordering you can get any lim inf and any lim sup you like (as long as the lim sup is at least as big as the lim inf).

@James: I just noticed your nice footnote about turning hot and cold taps in a shower. Nice analogy.

Sriharsha,

Good question. Notice that the first series begins with 1 -1/2 + 1/3 - leading to

0.6931…, whereas the second series begins with 1 + 1/3 - 1/2 + … leading to

1.0397… It is not only the order of the sumands that is changed, but also which terms are subtracted and which terms are added.

1)We have developed a lot of work in series summation and many theorems have been proved. So, Zenon's paradox is a typical summation example now.

2)There exist a gradient in infinity: some infinities are more dense than other infinities, to say it without to be 'hit' by mathematicians. There exist an increasingly cardinality of infinities and that is opening even cosmological questions (Why should the universe be less complex than a science (Mathematics) created by a small subset of it, ie from human beings?)

A big issue:

"Dedekind (1888):A set is infinite if and only if it is equipollent with some proper subset of itself."

If we use that result we can argue:

"Universe is multi-infinite since it is equipollent with the mathematical description of it".

The question is this: What kind of Universe? The material one that we can understand or the whole set of universes, material and immaterial?

Nice question.

Euler's paradoxical series is a nice example.

1, -1/2, +1/3, -1/4, ...

Add the first n terms of the series, you get some number depending on n, and as n gets larger it gets closer and closer to log(2).

If you add only the first n positive terms you get a sum which converges to +infinity as n -> infinity.

If you add only the first n negative terms you get a sum which converges to -infinity as n -> infinity.

By adding more and more negative terms, and more and more positive terms, but taking different numbers of positive and negative terms, you can arrange that the total converges to anything you like between - infinity and + infinity, by taking different balances between numbers of positive and negative terms.

For instance, with the ordering

1, 1/3, -1/2, 1/4, -1/5 ...

you always have either two or three more positive terms than negative terms.

I think that once you realize that the sum of the positive terms is infinite and the sum of negative terms is negative infinite, the paradox kind of disappears. There is clearly a huge reservoir of different possibilities, determined by choosing different balances between negative and positive terms.

There is of course a theorem which says that the limit of the series is the same under any reordering of the terms if and only if at least one of the two series (sum of positive terms; sum of negative terms) is finite.

If both positive and negative series diverge but the individual terms themselves converge to zero, then by suitable reordering you can get any lim inf and any lim sup you like (as long as the lim sup is at least as big as the lim inf).

@James: I just noticed your nice footnote about turning hot and cold taps in a shower. Nice analogy.

@James: you wrote: "In the late 19th century, Richard Dedekind gave the first rigorous theory of R. He found that it is necessary to not use infinitesimals, because there aren’t any."

This is not quite accurate. (I just noticed that you do have a footnote to this effect, missed it at first reading).

Dedekind found that there weren't any infinitesimals in *his* rigorous theory of R, but his isn't the only possible rigorous theory of R (though of course everyone behaved as if this was the case, except for the physicists, for another 100 years or so).

In "non-standard analysis" one does have a rigorous theory of R in which infinitesimals not only exist but can also be used in an effective way to short-cut many proofs and definitions. For instance, integrals are no longer limits, but just sums... and derivatives are no longer limits, but just ratios...

However this has not caught on.

(When I say that non-standard analysis is a rigorous theory of R, I mean, just a rigorous as the conventional theory. Who knows when we next discover some loopholes in the conventional theory. It has happened before time after time again and I don't see a reason why this process should ever stop).

Imre Lakatos' book "Proofs and refutations" is very good on this stuff.

@Richard,

Excellent posts! Apparently, Dedekind focused on what he called infinite systems and simply infinite systems. Any infinite system has as a part a simply infinite system (see p. 38 of the attached paper). For Dedekind and Bolzano, a principal task is to prove that there is such an entity called an infinite system.

@Richard,

Many thanks for pointing out Imre Lakatos's book (I will try to find a copy).

@James, I think you'll like Lakatos!

I loved this quotation of Hilbert's

“No one can expel us from the paradise Cantor has created.”

In the PDF file is mentioned a joke about a mathematician and an engineer at a dancing party.

Here goes

An engineer, and a mathematicia went to a dance. Shyly they positioned themselves against a wall where they had a good view of the dance.

The mathematician sighed heavily and said “I wish I could go ask one of those ladies sitting at that table over there to dance with me, but it is impossible.”

“Why is that?” asked engineer.

“If I go halfway over to the table, I will still have halfway to go” replied the Mathematician.

“Yes” Said the engineer.

“Then if I cover half the remaining distance I will still have a quarter of the way to go” Said the mathematician.

“Yes” Replied the engineer.

The mathematician continued “I can then cover half the remaining distance, but a 16th of the distance remains. Everytime you cover half the distance to the table a small but calculatable amount of distance remains. So it impossible for me to go over there and ask for a dance. To bad for us”- when the Engineer got up and walked over to the table.

The mathematician watched in amazement as the engineer asked a particularly attractive young lady to dance, proceeded to dance with her, gave her a lingering kiss, and then came back to their place on the wall.

“How did you do that?” asked the mathematiciant in awe.

“Although you were correct, I calculated that I would be able to get close enough for any purpose I could think of”.

"The known is finite, the unknown infinite; intellectually we stand on an island in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land." (Thomas H. Huxley)

The set of all natural numbers is our first infinite set, and it has a comfortable feel about it. Among infinite sets, the natural number seem the most natural. By examining this and related collections of numbers, we develop a better and more precise idea of infinity.

The fundamental idea in the study of infinity is that two collections have the same size if there is a one-to-one correspondence between the members of one collection and the members of the other collection. This compelling concept of comparing sizes via one-to-one correspondence is the rock on which the whole study of infinity is built.

Qefsere, you write: The set of all natural numbers is our first infinite set, and it has a comfortable feel about it.

However, not everyone agrees with Cantor's view of infinity relative to the natural numbers. The following things have been observed

Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[6] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[57] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[6] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.

For more about this, see

http://en.wikipedia.org/wiki/Georg_Cantor

And for an overview of the history of infinity, see the attached pdf file.

The references from the previous post are given below:

[6] J.W. Dauben, George Cantor: His mathematics and philosophy of the infinite, Harvard University Press, Boston, 1979.

[57] E. Snapper, The three crises in mathematics: Logicism, Intuitionism and Formalism, Mathematics Magazine 524, 1979, 207-216.

As it is well known potential infinity is used by the Greeks up to Gauss. The Cantor base his set theory on actual infinity. To day there is at least one example of set theory, that uses potential infinite and it is based on Husserl phenomenology. This is Vopenka's Alternative Set Theory.

@Demetris Christopoulos: You write: A big issue:

"Dedekind (1888):A set is infinite if and only if it is equipollent with some proper subset of itself." If we use that result we can argue:

"Universe is multi-infinite since it is equipollent with the mathematical description of it".

Whoops! Not quite. A mathematical description of the universe is not a proper subset of the universe.

There is also the meaning of Dedekind's use of the term "equipollent" to consider. In mathematics, two sets are equipollent, provided there is a 1-1 correspondence (bijection) between the two sets. See

G.H. Moore, Zemelo's Axiom of Choice: Its Origin and Development, Springer, 1982.

Dear James, my logic is a projectional one: Since the subset (math theory) is so complex, then the superset (universe or what ever) has to be at least more complex than a creature that some intelligent beings of it have created. Probably this view cannot be fitted with Dedekind's results. Can you provide me a proper mathematical theory that can support such a view? Thank you.

I would like to add a remark. For potential infinity we may write the conceptual equation:

infinity=fineteness+vagueness (or fuzziness)

Thus one of the characteristics of a non-Cantorian infinity is fuzziness.

Costras,

Your conceptual equation needs some explanation. Here are questions to consider:

1. How does the combination of finiteness and fuzziness yield (lead to) infinity?

2. What is your interpretation of fuzziness relative to non-Cantorian infinity?

@James Peters: What I mean is this: When we say, 1.2.3,... these three dots can not be defined rigorously. What remains is 1,2,3,...,n,... which means that the three dots at the end is just a vagueness assignment. The important think here is to connect the potential infinity, or non-cantorial infinity with vagueness or fuzziness. Suppose you are observing with your naked eyes 4 students. Then you see then through a very disturbing glasses. Then you have the "phenomenon" of continuity! You see the four students as a continuum. Thus the vagueness of discriminating each student leads you to continuum, i.e. to infinity!

Costas,

With the three dots …, you have an interesting idea. It is still not clear how you connect potential infinity with fuzziness, but I can see what you mean by vagueness. The problem with what you are suggesting reduces to the meaning of

1. infinity. By writing "potential infinity", this suggests that we already know what infinity is. Perhaps you will agree that infinity itself needs to be defined before we can talk about potential infinity.

2. fuzziness. Unfortunately, your use of the term makes it synonymous with the term "vagueness". But fuzziness is defined in terms of graded members. Something (call it x) is fuzzy, provided the membership of x in a set A is a value between 0 and 1. Hence, the fuzziness of x is not vague, since the graded membership of x in A is precisely known as some m(x) value in [0, 1], where m: A --> [0,1] is a membership function. I assume that you mean by "vagueness" the colloquial sense of the term.

See the attachéd pdf file for a very detailed view of fuzziness.

@James: You are right about fuzziness vs vagueness. I use it in a rather colloquial sense . I would not agree with you, that we have first to define a concept and then talk about it. The concept of infinity is already in the conceptual background of any human and especially the generic human. Definitions, and/or discussions have to capture the concept in a rather clear way. For me cognition/perception is before language and rigorous definition.

In our example of potential inference, it is clear that e.g natural numbers should be written as, 0,1,2,3,... and the actual natural numbers as N:={0,1,2,3,....}. Thus potential infinity is an open series of numbers whereas actual is a closed one. For kinds of infinity, the three dots, are not definable! From this I infer that dots could be replaced by a finite set (0,1,2,3,...n} plus some undetermined term.

I would like to note, in passing, that are nonstandard models of natural numbers, such that, big numbers are playing the role of infinite large numbers. To let you understand what I mean let us have a series of white cards and some white colour. Start colouring the cards with the colour, and every time you put a drop of black colour, in such a way that you can not discern one card from the next one. At some point you see just a black cart. Then all the white carts are the natural numbers, and they satisfy also Peano axioms. The black numbers are the infinite numbers! a similar model you can find in Vopenka's Alternative Set Theory.

As you can see, in this model the "infinity" might be interpreted as something "incomprehensible". From this point of view infinity is a kind of vagueness.

There are also a lot of topos theoretic models of natural numbers. Trying to catch the substance of infinity is a rather complex problem. The only possibility that we have, is to study as much as possible of these different models, then we may start to philosophise about "infinity" and finally we may be able to proceed with a definition that encompasses all faces of "infinity".

I hope that I not confuse you, with this long note.

Dear Costas, I found your example interesting because I hadn't thought about infinity as emerging from vagueness (to be strict). But, there exists a problem here and that's the 'perceived reality' concept: From your example the infinity obviously do not exist (only 4 students exist), so how can this process be used to create the 'true' infinity? How can we get rid of false infinity?

@Demetris: The example I gave is just coming out of "vague predicates". In addition there are many non-cantorian types of infinity. Frankly I do not understand what do you mean by "true and false infinity". I insist that in order to understand deeply the concept of "infinity" we should study in detail various non-Cantorian models of infinity. I believe that the hart of the matter of non-Cantorian infinity lies in the concept of vagueness. In fact the negation of the excluded middle leeds essentially to many-valued logic and this in postmodern mathematics, where the grey, vague, etc. is the mode. I may give you another example based on familiarity: We are more familiar with 1 rather than 2, rather than 3, ....,rather than 2^25, rather than n. This introduces a kind of vagueness based on "familiarity". from this perspective 2^25 is not a finite number but it is infinite! This kind of thinking introduces a generic "observer", and the "infinity" that he perceives is just what we are talking about. You see that, "pluralism", "man-valued logic", "subjectivity", etc. are characteristics of postmodern condition.

Dear Costas, I understood your logic. When I wrote about 'false infinity' I meant that the continuum of the 4 students are not so, i.e. it is a false observation due to our limited optical sense. So, as infinity I'd prefer to accept a concept that could be defined outside the relativistic properties of our material senses.

Dear Demetris, sooner or later we have to accept things we learn that it is not true! Mathematics for the physical nature are OK, but we need something else, or some kind of generalisation of the classical mathematics. These opinions os course are very much subjective!

Costas, you write: As you can see, in this model the "infinity" might be interpreted as something "incomprehensible". From this point of view infinity is a kind of vagueness.

Your examples and explanations are cogent up to a point. However, identifying infinity with incomprehensible is stretching things a bit, since you righly shy away from defining the term infinity. There is still the open question about how the current view of infinity differs from infinity in classical Greek mathematics. It would seem that infinity in classical Greek mathematics was synonymous with "unboundedness". For example, this unboundedness can be seen Euclid's observation about the number of prime numbers. In other words, the meaning of infinity comes from the idea of "not finite" and "unlimited". In effect, one can discern in classical Greek mathematics a definite view of infinity . And this definite view of infinity carries over in current mathematics. But now we benefit Cantor's discovery of orders of infinity (see the end of the attached pdf for a view of different kinds of infinity).

So this brings us to your second observation, namely, "infinity is a kind of vagueness". Perhaps now you can see that the notion of infinity (either currently or in classical Greek mathematics) is definite, not vague.

@James: Dear James, what you attached is just the well known treatment of Cantorian actual infinite. What we are looking at is rather non-Cantorian treatments of potential infinite.

So e.g look at: Dov M. Gabbay, Akihiro Kanamori, John Woods (Editors) Handbook of the History of Logic- Volume 6- Sets and Extensions in the Twentieth Century 2012.

I am especially interested in nonstandard analysis based extensions. I divide the existed nonstandard theories into "extensional" which is Robinson theory, where infinitesimals posses an infinitesimal extension, and the "intentional" one where the infinitesimal is perceived as such by a generic observer. Internal Set theory of Nelson and an extension such as Vopenka/s Alternative Set theory are such models.

Note also that Semisets can be used to represent sets with imprecise boundaries.

See e.g.

You can contact also:

http://plato.stanford.edu/entries/settheory-alternative/#SetTheForNonAna

http://en.wikipedia.org/wiki/Alternative_set_theory

http://en.wikipedia.org/wiki/Vop%C4%9Bnka

http://en.wikipedia.org/wiki/Semiset

Dear Costas,

Good post! I will check the references you have given. Very interesting!

We are still faced with two issues raised by your earlier post, namely,

1. Is any form of infinity synonymous with incomprehensibleness?

2. Is any form of infinity some form of vagueness?

@James: Dear James, of course not any form of infinity is synonymous with incomprehensibleness. Cantorian actual infinity is developed by generalising properteis of finite sets to actual infinity. Of course the question remains: how comprehensible is Cantorian concepts of infinity? Surely this infinity is not introduced as a form of incomprehensibleness. Neither this kind of infinity is some form of vagueness.

What I say is that non-Cantorian infinite is related very much to vagueness. you can see this by seeing Alternative Set Theory of Vopenka. In this theory everything is "finite" from Cantor’s point of view. A set is finite iff each subset of it is a set, i.e finite means hereditary finite. Classes that are not finite are called infinite. Thus an infinite sets is a class that contains a “semiset”. Since semisets are used to define sets with vague boundaries, it is clear that “infinite” here” means vagueness!

Dear Costas,

Your last post and the previous post with the references to Vopenka's work on semisets, are definitely relevant and important additions for this thread. Before tackling the question of infinity and vagueness, I offer here a summary of the basic axioms for Vopenka's semiset theory.

1. Identical classes have the same elements.

2. Some classes are elements of other classes. Such classes are called sets.

3. A semiset is a subclass of a set.

4. For any two sets, there is a pair set.

5. Any reasonable property determines a class, namely, the class of all sets

having the property.

6. There exists an infinite set.

7. Given a 1-1 correspondence between sets and semisets, the number of sets

is small, provided the number of semisets is small. Reasonable semisets are

those semisets determined by reasonable properties.

8. Reasonable semisets are sets.

In addition, there is Vopenka's axiom of infinity:

There exists a set X such that the empty set belongs to X

and for all y in X implies {y} is a member of X.

It is clear from the axioms that semisets do not have vague boundaries. In addition, it is clear from Axiom 6 that an infinite set is comprehensible. And from the axiom of infinity, it seems that infinity is not synonymous with vagueness.

To help carry this discussion forward, see

http://sistemas.fciencias.unam.mx/~lokylog/images/stories/Alexandria/Teoria%20de%20Conjuntos%20Avanzados/ebooksclub.org__The_Theory_of_Semisets__Studies_in_Logic_and_the_Foundations_of_Mathematics_.pdf

The pdf file contains Vopenka's Theory of Semisets (a 13 MB file). See pages 6-7.

Dear James, Thanks for the book, which already had. The problem now is the difficulty to understand the axioms and the concept of class. In Mathematics in the Alternative Set Theory, p.11 we find: "We shall deal with the phenomenon of infinity in accordance with our experiance, i.e., as a phenomenon involed in the observation of large, incomprehensible sets."

Furthermore, in http://en.wikipedia.org/wiki/Semiset we find: "Semisets can be used to represent sets with imprecise boundaries. Vilém Novák (1984) studied approximation of semisets by fuzzy sets, which are often more suitable for practical applications of the modeling of imprecision." You can check the proof of Novak to see the details.

Anyway, these are the thinks I have to say for the problem at hand. If you still are not convinced, thats OK with me.

In order to understand Alternative Set Theory, a non-Cantorian set Theory, one should look at:

1. P. Vopenka, Mathematics in the Alternative Set Theory. Taubner, 1979.

2. P. Vopenka, THE PHILOSOPHICAL FOUNDATIONS OF ALTERNATIVE SET THEORY, International Journal of General Systems Volume 20, Issue 1, 1991

3. Vopenka-Trlifajova-Alternative Set Theory, In-Floudas-Pardalos, Encyclopedia of optimization, pp. 73-77 2008

Which I attached:

The 3. paper.

Dear Costas,

Very good posts! You are making good points. Thanks to your comments, there are new interesting questions about infinity to consider.

1. Is an approximation of a semiset with fuzzy sets beneficial? This question raises an important issue: When is it appropriate to use fuzzy sets to clarify and reason about a set?

2. Which of Vopenka's axioms require proof? In other words, is each of Vopenka's axioms assumable but not provable? Of particular interest is Vopenka's axiom of infinity vis-a-vis Cantor's view of infinity.

3. In what sense is alternative set theory non-Cantorian?

Dear James,

Approximation of semisets by fuzzy sets, makes them more applicable. Thinking in general might require to leave semiset untouched.

Usually, we do not prove axioms. However the axioms and all philosophy of AST is based on Husserl phenomenology. Actually AST is a generalisation Von Neumann–Bernays–Gödel set theory, by relaxing the axiom of separation and introducing semisets.

Cantorian set theory is something close to a kind of Platonism: white-black. Everything is crisp. Non-Cantorian on the other introduces a set theory close to our experience of outer real world. A phenomenology of "finite sets". Through the concept of semiset introduces a kind of vagueness, which identifies AST as non-Cantorian. We should be careful by leaving "infinity" for the Cantorian infinite and using "unlimited" for the non-Cantorian infinite. And now I return to my initial conceptual equation:

non-cantorian infinity= finite+vagueness

meaning a semiset, i.e. a finite set which includes a class.

Dear Costas,

Good post! You have raised a number of important issues, one of them surprising, namely, alternative set theory (AST) being based on Husserl phenomenology. I still have not had a chance to follow up on what you wrote earlier about AST. Notice the dilemma I face in reading what you have written about the meaning of semiset and Vopenka's axiom 6:

6. There exists an infinite set.

Perhaps you will agree that your non-cantorian equation needs revision. Consider, for example,

non-cantorian infinity = (finite or infinite) + vagueness

And I still have difficulty with your suggested approximation of semisets via fuzzy sets. If anything, this fuzzy approximation of semisets muddies the waters and makes something that is crystal clear not so clear. I suggest that it may be more interesting to consider the nearness of semisets by endowing a space containing semisets with either a spatial or descriptive proximity relation. I suggest this because it paves the way to some interesting theorems and applications of semisets.

Dear James,

Vopenka has three cardinalities: finite, countable, and continuum. actually these are the cardinalities which the everyday mathematician is using. The axiom "There exists an infinite set" is needed since otherwise semisets could not otherwise exist in a formal theory. Countability is used only to determine a "horizon" of observation. Continuum lies beyond horizon. Inside the horizon we always have finite sets. Infinite sets are agsin finite sets which include a semiset. Actually a semiset imposes a level of reality. If you have three material balls, then these balls have a microscopic reality, with electrons and particles that make up a huge incompressible set, i.e. a semiset. Since AST is a kind of nonstandard analysis closer to Nelson's IST, which regard "intentional" rather than Robinsons's which is extensional. In nonstandard analysis we have at least two levels of reality. May be you can compehend "semiset" by considering external sets in NSA.

There is a need for writing a paper of a books which will compaire all these and finally will introduce a unified notation.

My conceptual question is OK for me! As for the approximation of semisets by fuzzy sets, this was an idea of V. Novak, and if do not like it, I cannot help it.

Dear Costas,

Many thanks for your interesting post! It is Vopenka's idea of an infinite set that sets him apart from Cantor. The focus of Cantor's work is on the cardinality of the natural numbers, leading to orders of infinity.

By contrast, Vopenka shifts the focus from the cardinality of an infinite set to an infinite set itself. In other words, Vopenka shifts the focus from infinite magnitudes to infinite sets.

Dear Costas,

There is something that I forgot to mention in my previous post. Let X be an infinite set endowed with a descriptive proximity relation and let A be a finite subset of X. Further, let Y be a subset of R^n. It is then possible to define a continuous homeomorphism on A into Y. The homeomorphic mapping f on A into Y is defined in term so feature vectors, each contain n real values representing features of members of A that describe each a in A, so that f(a) belongs to R^n. Descriptive proximity is important here because we can let each member of A be a descriptively adherent point (i.e., each point in A has a unique description, so the inverse mapping can be defined). In effect, the set A is a manifold in relation to Y.

What may be interesting to consider is a manifold that is an infinite set.

Dear James, thanks for your interesting remarks! Your last remark is really interesting. As AST is a form of nonstandard analysis, using either one of these tools, we may reduce the infinite case to "finite" or hyperfinite to be precise.

I would like to comment on the sentence "Cantorian actual infinity is developed by generalising properteis of finite sets to actual infinity" in an earlier post.

In the first place, infinity being a denial of finiteness, there is little that can be generalized from one concept to the other. Perhaps "adaptation" is a better term than "generalization".

Secondly, it seems to me that (generally speaking) a mathematical discourse in an an infinite set quite often makes no explicit appeal to infinity. Instead, infinity is just lurking behind the scene as a logical consequence of the data at hand. In many cases there is not even a need to draw that conclusion explicitly.

To put it boldly, infinity is much more often an (implicit) consequence than the opposite case, where plain consequences of infinity are drawn. Thus, there are relatively few "properties of infinity" versus lots of "infinitary aspects of properties".

At the positive side, I should say that I appreciate the friendly discussions in this thread and the interesting references.

The contradictory “potential infinity” and “actual infinity” concepts have being accompanying us with infinity related paradoxes and a long-drawn-out and ceaseless “potential infinity--actual infinity” debate since antiquity. But can we really have two, three, four or even more contradictory infinity concepts with different natures in our science? How do the contradictory infinity concepts with different natures exist theoretically and operationally in our present science? These are the questions unavoidable for anyone who studies and works in the field of infinity. If the contradictory “potential infinity” and “actual infinity” concepts are wrong (unscientific) their relating logics are surely wrong (unscientific) and many things relating to infinity in our mathematics (in analysis and set theory) are chained wrong (unscientific).

Do we have “potential infinity number, potential infinity set, …” and “actual infinity number, actual infinity set,…” in our mathematics? Is "x---->0" number forms treated by limit theory potential infinity numbers or actual infinity numbers (number of no number)?

Dear Geng, 

I would suggest to read some previous post of mine. To help you I collect those posts. May be might be of some help to you.

"

As it is well known potential infinity is used by the Greeks up to Gauss. The Cantor base his set theory on actual infinity. To day there is at least one example of set theory, that uses potential infinite and it is based on Husserl phenomenology. This is Vopenka's Alternative Set Theory.

I would like to add a remark. For potential infinity we may write the conceptual equation:

infinity=fineteness+vagueness (or fuzziness)

Thus one of the characteristics of a non-Cantorian infinity is fuzziness.

What I mean is this: When we say, 1.2.3,... these three dots can not be defined rigorously. What remains is 1,2,3,...,n,... which means that the three dots at the end is just a vagueness assignment. The important think here is to connect the potential infinity, or non-cantorial infinity with vagueness or fuzziness. Suppose you are observing with your naked eyes 4 students. Then you see then through a very disturbing glasses. Then you have the "phenomenon" of continuity! You see the four students as a continuum. Thus the vagueness of discriminating each student leads you to continuum, i.e. to infinity!

I insist that in order to understand deeply the concept of "infinity" we should study in detail various non-Cantorian models of infinity. I believe that the hart of the matter of non-Cantorian infinity lies in the concept of vagueness. In fact the negation of the excluded middle leeds essentially to many-valued logic and this in postmodern mathematics, where the grey, vague, etc. is the mode. I may give you another example based on familiarity: We are more familiar with 1 rather than 2, rather than 3, ....,rather than 2^25, rather than n. This introduces a kind of vagueness based on "familiarity". from this perspective 2^25 is not a finite number but it is infinite! This kind of thinking introduces a generic "observer", and the "infinity" that he perceives is just what we are talking about. You see that, "pluralism", "man-valued logic", "subjectivity", etc. are characteristics of postmodern condition.

What we are looking at is rather non-Cantorian treatments of potential infinite.

So e.g look at: Dov M. Gabbay, Akihiro Kanamori, John Woods (Editors) Handbook of the History of Logic- Volume 6- Sets and Extensions in the Twentieth Century 2012.

I am especially interested in nonstandard analysis based extensions. I divide the existed nonstandard theories into "extensional" which is Robinson theory, where infinitesimals posses an infinitesimal extension, and the "intentional" one where the infinitesimal is perceived as such by a generic observer. Internal Set theory of Nelson and an extension such as Vopenka/s Alternative Set theory are such models.

Note also that Semisets can be used to represent sets with imprecise boundaries.

See e.g.

You can contact also:

http://plato.stanford.edu/entries/settheory-alternative/#SetTheForNonAna

http://en.wikipedia.org/wiki/Alternative_set_theory

http://en.wikipedia.org/wiki/Vop%C4%9Bnka

http://en.wikipedia.org/wiki/Semiset

ALTERNATIVE SET THEORIES,In- Dov M....in the Twentieth Century 2012.pdf"

Dear Costas,

Many thanks for collecting together you previous very helpful posts.   That does help move the discussion forward.    Hopefully, this will offer some food for thought in terms of @Geng Ouyang's observations.

Dear Costas,

Many thanks for your idea.

According to my studies, whenever contradictory “potential infinity” and “actual infinity” concepts exist simultaneously in our science (including mathematics of cause), none is able, possible to hold only “potential infinity” but ran away from “actual infinity”, vice versa because the two concepts coexist. This conclusion was proved by my second 15 minutes presentation in 2014 ICM in Seoul (has been uploaded on RG).

Dear Geng,

Potential infinity, has a life from the ancient Greeks, to Gauss. After G. Cantor, actual infinity was the dominating concept, paying no attention at all to potential infinity. In our days, there is Nelson's Internal Set Theory (IST), which some e.g. Vopenka has been given a non-formalistic account, and interpreted concept of IST in terms of potential infinite. Please read the paper by Vopenka. 

I disagree that the two kinds of infinity "exist simultaneously in our science (including mathematics of cause)". Your "none is able, possible to hold only “potential infinity” but ran away from “actual infinity”, vice versa because the two concepts coexist" also do not hold. Vopenka's "Alterantive Set Theory" is based on potential infinity only, and can codified all mathematics. 

I will read your presentation in 2014 ICM in seoul and I' ll be back.

Anyway thank you for this exchange of ideas. 

Dear Costas,

Thank you for your patient and kindness. My work bases on following three points:

1, the defects disclosed by Zeno’s Paradox “Achilles and Tortoise” are still unsolved（countless papers can be found ）.

2, Berkeley’s Paradox coming along with Zeno’s Paradox in the exactly same environment : infinitude conception（such as “potential infinity " and “actual infinity "）, infinitude number conception（such as infinitude number forms, infinitude number system）, treating theories and techniques of the infinitude number forms（such as limit theory）; so, in present traditional infinitude system, these twins are with the same fate, if Zeno’s Paradox can not be solved, so is not Berkeley’s Paradox and vice versa.

3, with the confused definitions of “potential infinitude” and “actual infinitude” (long-drawn-out and ceaseless “potential infinity--actual infinity” debate since antiquity proves this conclusion), many people working in the field of infinitude are unable, impossible to know clearly what mathematical things they are facing to and treating-------“potential infinitude” or “actual infinitude”.

Do you agree with the above three points? I sincerely hope you can give me you opinions if not.

If the above three points are all wrong then my works are all wrong.

Best Regards

Dear Geng ,

By studying and focus on paradoxes is a great  idea. Paradoxes usually hide treasures, if one can understand and resolve paradoxes.

Starting from  Zeno’s Paradox, one can resolve it using infinitesimals. There are two types of infinitesimals: Robinsonian, which are based on actual infinity and Nelson's or Vopenka's, based on potential infinity. 

By putting all these paradoxes together (mathematical, philosophical etc.) I think it is not a good idea. I did not find your presentation good and convincing. My suggestion is to give a try to rewrite your basic paper following some rigour accepted by the mathematical community. Your ICM paper does not give a positive sense!

If I can be of any help, please do not hesitate to ask. 

(quote) The concept of infinity is already in the conceptual background of any human and especially the generic human. Definitions, and/or discussions have to capture the concept in a rather clear way. For me cognition/perception is before language and rigorous definition. (end quote)

@Costas: You are an "emotional mathematician". That's fine. But the end product -- as far as we are concerned here -- is mathematics, which is blind and deaf to emotions. Your quote would do well for a didactic remark as a teacher, or as a writer of a nice mathematical paper. I greatly appreciate your description of white cards slowly turning black to motivate infinity as finite + something fuzzy. But...

As a sober mathematician (taking joy in colorful descriptions!) I must observe that mathematics is ultimately about precise communication by means of unambiguous statements. The essence of the natural numbers, stripped from didactic niceties, is a claimed "set" containing an element 0 and with a successor operator having certain clearly stated properties.There is no triple of dots suggesting "etcetera". The word "infinity" does not occur and is not even (directly) suggested.  It is only a by-product (consequence) of the selected properties and that's quite usual in axiomatic definitions.

This is one of the (many) points I would like to make. People seem to think that if a set is infinite, you cannot collect it because the collecting process is endless.

Why on earth should you collect the elements? I assume you do not have the complete collection at home of all (post)stamps that were ever produced. But if I show you a colored piece of paper, you are still able to decide whether it belongs to the proposed collection by consulting a good catalogue.

And indeed, collections can be handled without explicit collecting. Either the set is named in the axioms (eg, empty set, set of natural numbers) or it is build by one of the set-building operators. One of them is {x | x in A and P(x) } (where "P" stands for a predicate). To decode membership, you must decide on "x in A" and on "P(x)". Much as with the stamps collection! Do you see a process of collecting things that does not end?

Corrected with axioms, set theory is not a priori concerned with "actual infinity" or even "potential infinity" (which are associated with the avoidable act of collecting). Borrowing a term from Artificial Intelligence ("lazy construction"), I would say that set theory knows only "lazy infinity".

Cantors proof that a set is "smaller" than its power set works just the same for finite sets. (You are supposed not to know that n < 2^n.) Intuitionists won't like it because it reasons by contradiction. Where is the problem with infinity?

Surely, infinity is the ultimate reason that certain problems are undecidable. Hilbert's dream of a complete and decidable mathematics is an illusion. Mathematics is our most precise instrument, and is called "unreasonably effective" by physicists. But hey, we are humans and there is nothing perfect under the sun. 

Dear Marcel, 

Thanks for the attribute  "emotional mathematician"! I would prefer sensual mathematician! Any way, I must tell you that what I try to say by the cards example is to explicate what a vague predicate is. Actually the example is not mine. It is R. Parikh who introduce the concept of vague predicate! 

Then you say "As a sober mathematician (taking joy in colorful descriptions!) I must observe that mathematics is ultimately about precise communication by means of unambiguous statements." I do not like that staff about "soberness", and we should discriminate "mathematical thinking" from "mathematical communication". For example when the thing is about "vague predicates" the colourful descriptions do not preclude rigour.

I think our difference is that your talking about mathematical communication and I am talking about Mathematica thinking.

I imagine that you will reject Vopenka's ideas, as well, on the basis of "soberness". I am convinced that there are "sober mathematicians" who do not know what there are talking about. They just like to play with symbols! 

For your opinions about actual and potential infinity I disagree completely! Cantorian set theory, could not proceed to ω, ω+1, ω+2, ... if we could not complete  ω, and this is actual infinity! I will ask you to read Vopenka's attached article and there you will learn in a sober way, what is actual and what is potential.

i have not checked of what field you are coming, but I sure you do not care about foundations and philosophy of mathematics.

Dear Costas,

Thank you ever so much for your kindness and helpful mind. I feel the art of a good teacher.

But I am deeply convinced that by putting all these infinity related paradoxes together (mathematical, philosophical etc.) can let us understand better the inside infinity related fundamental defects which bound them together and it is the only way to see through the nature of these infinity related paradoxes because the infinity concepts and their relating logics are the same. This is the biggest difference on working train of thought between me and others, this is a break through.

Could you please chose one of my papers you think possible to be reformed and we cooperate to rewrite it?

I will be back 2 days later.

Dear Costas,

As a mathematician, I am probably not much different from you. I like beauty in proofs so much that I sometimes overlook an "ugly" alternative. When I think mathematics, my intuition and imagination are fully operational. In most of my research, the major result came as a sudden breakthrough (often with adjustments afterwards) or as an unintended side effect of efforts. Mathematicians working as a constantly buzzing logic machine are rare, I think.

I do not reject Vopenka's ideas: it is a precise axiom system. I have some experience with non-standard models in logic, so I am familiar with the idea of having a class within a set. I may  well return to the subject sooner or later, but at this moment I wish to stick to  traditional set theory.

Our apparent disagreement comes from the fact (which you noticed too) that we are talking about two different things: mathematics itself and "doing mathematics". As a "do-er"  of math I have sympathy for most of what you say. But when this thread asks about changing views on infinity, one should at least have a look at  mathematics itself, the "core subjects" at which our daily activities are directed. Just as our vision is flooded with mental constructs, we project intuitions on this mathematical core to get the feeling of it (with great success, apparently):

When we consider a converging sequence, we see a row of entities running into a horizon, moving to a limit. We consider maps as operators which really map something, we think of continuity as moving a pencil over a paper without interruption. Most names in mathematics reflect some view or intuition: collection, measure, probability, distance, vector, larger/smaller, ... The list of intuitions and mental presentations is endless and has much personal variation. It is no doubt the reason that we are good at mathematics, because psychological experiments indicate that pure logic thinking is not really our habitat (we guess cause from effect, we do much wishful thinking and tend to neglect what we don't like.

However, the "core of mathematics" is totally static and without emotions; there are no processes running, beautiful ideas may turn out wrong, and logic errors don't make the heavens fall down. A sequence doesn't run to its limit, a map doesn't move anything around. And most importantly, every piece of mathematics represents a finite amount of information. It would even be preposterous to think it otherwise: we have no magic powers for infinity. We think of the natural numbers as "an infinity". What we really use is the freedom to take a larger number if needed. Our sensitivity to boundaries makes us eager for infinity.

Giving a name (say, ω) to the natural number system is like attaching a label to a bunch of properties which you "activate" when using the map with label ω. There is no obligation to have all natural numbers listed: writing down ω, ω+1, ω+2, ... does not require ω to be "completed" first. This habit is probably inspired by the synonym "collection" for set.  Cantor held the "naive" idea that the mere thought of some "host of things" is enough to "have it collected in the mind". It needed axiomatic correction. Since then, there have been no real problems with standard set theory.

It is wise to know about the ground under our mathematical feet. We do not really involve in infinity; I stick to "lazy infinity".

Dear Marcel,

Let me take some of your points and comment on them.

“is mathematics, which is blind and deaf to emotions.”

I think that all your statement hide a certain philosophy that you seeing things. The above statement hides such a philosophy. There is however some philosophies, which are not deaf to emotions! Starting from “intentional” mathematics” that incorporates classical and intuisionistic mathematics (See, e.g [Stewart_Shapiro] Intentional Mathematics) and end up with intentional in Husserl’s sense, and intensional of how we see, how we hear, touch etc.

I greatly appreciate your description of white cards slowly turning black to motivate infinity as finite + something fuzzy. But...

The cards is an example give by ROHIT PARIKH THE PROBLEM OF VAGUE PREDICATES and I do not see any problem of using such example in order to make clear what we are talking about. The strict formal way hides the “conceptual background” and sometimes is comprehencible by only a few specialists. This kind of writing I hardly accept for writing papers, since a good paper is a very well understandable and conceptually clear paper. A related article is

Reyes, G. [1986] Synthetic Reasoning and Variable Sets. In Categories in Continuum Physics, edited by Lawvere and Schanuel. Springer LNM # 1174, 69-82.

In the above paper the objective is a “synthetic language” in which we can communicate our results to almost all scientist and we can do research as well. In this way we can be free from strict formal way of expressing math.

“…mathematics is ultimately about precise communication by means of unambiguous statements. The essence of the natural numbers, stripped from didactic niceties, is a claimed "set" containing an element 0 and with a successor operator having certain clearly stated properties.There is no triple of dots. The word "infinity" does not occur and is not even (directly) suggested.”

I really disagree with your statement above. From a foundational and philosophical point of view, we do not like to use a forma language to hide “under the carpet” all important questions. I may, if you like introduce the notation: 0,1,2,… for potential infinite and {0,1,2,…} for actual infinite. In set theory: The number 0 is defined to be the empty set, the set that has no elements.

“For any set x, one defines its successor Sx to be the set whose

elements are the elements of x together with x itself. A set X is said to

be inductive in case 0 is an element of X and whenever x is an element

of X then its successor Sx is also an element of X. The axiom of infinity

of ZFC asserts that there exists an inductive set. Then one proves that

there exists a unique smallest inductive set and one defines N, the set of

all numbers, to be this set.” So in set theory to go from 0,1,2,… to N:={o,1,2,…}

we use an axiom! And this makes Cantor Set theory a theory based on actual infinity. The higher ordinals we get by using unions of the pressending ones.

Finally you insist to use a formal language that hints the concept of “potential infinite” and the problem of collecting 0,1,2,.. into a set N.

People seem to think that if a set is infinite, you cannot collect it because the collecting process is endless.

Why on earth should you collect the elements? I assume you do not have the complete collection at home of all (post)stamps that were ever produced. Perhaps you have no stamps at all and know nothing about stamps. But if I show you a colored piece of paper, you are still able to decide whether it belongs to the proposed collection by consulting a good catalogue.

I show that you value Vopenka’s system of AST, since it is given axiomatically! But before one and up with the axioms a lot of conceptual work is needed! And I would like to see how an apartment has been constructed at all stages, rather that admire the finished result! As you see, to collect the elements 0,1,2,… into N, they do this axiomatically. But axioms hide some conceptual substance, and this is the most important for me! Furthermore the two infinities, potential and actual result into two different sets of natural numbers, potential results to a non-Cantorian set, whereas actual result to Cantorian N.

You prefer the formal expressions and that’s OK!

Corrected with axioms, set theory is not a priori concerned with "actual infinity" or even "potential infinity" (which are attitudes rather than properties or objects).

As for that, please have a look at the following:

E. Nelson Completed versus Incomplete Infinity in Arithmetic.

William C Waterhouse Gauss on infinity, Historia Mathematic Volume 6, Issue 4, November 1979, Pages 430–436

I haven't reacted yet on Costas' remark

"I [am] sure you do not care about foundations and philosophy of mathematics."

I do have a problem with the role of philosophy in the question of this thread. Since antiquity, throughout the middle ages, and deep into the 19th century, metaphysics has been a cherished subject of philosophy. Roughly speaking, at least the speculative aspects of it became disfavored around the time that set theory was introduced, with (among other things) a spectacular result of Cantor about different magnitudes of infinity. I hesitate to say this, but it seems to me that this offered an opportunity for philosophers to keep a foothold in speculative metaphysics.

My point is this: it looks like an amazing luck that Cantor's result on power sets can be derived correctly from a finite (and in fact very small) amount of information. Apparently, the disproportion between the numbers n and 2^n grows so explosively that the difference can even be seen "at infinity" from finite information. Other topics on infinity involve e.g.,"inaccessible cardinals" or the position of the continuum. This is the place to be for mathematicians who like undecidable statements. Does it sound surprising?

As mathematics --through its applications-- also takes a view of reality, philosophy certainly has something to say about its (human) interaction with the world and (human) belief systems.

However, most physicists take for granted that the universe is finite (both in size and granularity). Some don't. One may even wonder whether we will ever notice if it were infinite in one or both senses, but you never know with those clever physicists. For the time being, infinity is not part of accepted reality and hence a speculative topic in philosophy unless it restricts to the human factor. In "core mathematics" I notice only finiteness of information. In the parts of mathematics that encounter infinity, it is mostly a by-product of the assumptions at hand (e.g., a non-trivial connected Hausdorff space is infinite) and --if used explicitly-- it mostly does in a simple way as "a lack of boundary".

Dear Marcel,

Concerning philosophy I almost agree with you. But because of all these multitude of philosophies, does not mean that we should stop looking for a satisfied view. There should be a synthesis compatible with reality. I find Husserl very interesting for example. I presently write a review for AMS of the book: Danielle Macbeth, Realising Reason. Oxford 2014, and she combines a modification of Frege with the pragmatisms of Peirce, and finally arrives at an interesting view. Anyway, if a philosophy enlightens understanding of mathematics I like it, otherwise I am indifferent.

The kind of foundations I am interested you can have an idea by looking into my papers:

https://www.researchgate.net/publication/234059194_Structures_Points_and_Levels_of_Reality

https://www.researchgate.net/publication/237332784_Sets_Categories_and_Structuralism

Chapter Structures, Points and Levels of Reality

Article Sets, Categories and Structuralism

Costas,  I enjoy this dispute but we must be careful not to bore other participants in this thread. I will take a look at your references (I already did some reading around), which may result in a long silence. Before that, let me be explicit on a few points where I notice a potential misunderstanding.

1. What I loosely call "core mathematics" is not mathematics as it should be done , or as it should be communicated. It is the "ugly beast" that phrases everything explicitly in the set-theoretic language of (say) ZFC and the formal language of standard first order logic. Most mathematicians (including me) are satisfied when their (verbal) proofs are sufficiently detailed to be confident about correctness. We don't like to face the beast. Then why do I recommend to face it closely, just for once?

We consider a set as the entirety of its members and we are encouraged in this attitude by the axiom of extensionality. For (small) finite sets, given by explicit listing, this ontology is perfect. When two sets are defined via a property, one rather tends to prove that the properties are equivalent. That suggests a different ontology, even for finite sets. For complicated, large, or infinite sets, it is the most pragmatic one. Things are as they behave. As mathematics behaves only through our actions, one must move to a context where our actions are most elementary to get the purest picture: a set is an amount of information to be processed by logic operations (mind the axiom schema of specification). This creates a kind of fuzzyness like the one you are looking for and at no additional cost. As a child, I once counted to 1000 just for the fun of "having them all". When I finished  I realized that I just wasted my time. Please reread my example of "labeled maps at an office" in my second previous post.

2. Within one generation I expect there be software that does mathematical reasoning at about our level and technically even beyond, owing to massive parallelism. It will do so without emotions and at a somewhat more formal level. It would be a waste of efforts to let it think or grasp meanings in human language only. Rather, it will reduce the meaning of a concept to the set of possible actions suggested by it. For a basic concept, this amounts to its definition: prove an equivalence, prove an implication, etc.

If you need another reason why formalism can be useful, you may take a very brief look at my paper "Theories with the Independence Property" (available on RG). It is about a rather bizarre property of many, even quite common, logical theories, overlooked by mathematicians but not by computer scientists from AI. The reason why it is bizarre is explained informally in the first few sentences of the introduction.

Oh, there is one funny thing I like you to face. Suppose all of mathematics, done since the introduction of ZFC and formal logical, would be collected on a single computer. I mean: every concept, every result, every proof that can be translated into ZFC (in principle, anything). Even more: every single number and every single set  ever used, considered, or obtained  by a mathematician must be included. It must be made available in the language of logic and sets in a fully informative way. Infinite sequences of digits must be described, The number Pi must be defined and its existence as a real mumber should be proved. Named ordinals must be defined.

My civilized guess is that this takes some thousands of terabytes (just consider that nearly all of mathematics is available multiple times in the joined computer memories on earth).

A century of mathematics, able to communicate about infinity, resulting in merely 10^15 bytes of communication...

Dear Marcel,

Although a favour  Category Theory, I appreciate Set Theory and First Order Logic. What I do not like is that Set Theory and First Order Logic, became a monopoly that they do not stand oher voices. Personally I regard Set Theory and First Order Logic as the mathematics of the left hemisphere, whereas Category Theory, a non-analytical theory (a holistic theory) is compatible with right hemisphere. If this is so, then we would like the synthetic mathematics of the corpus calosum. I regard algebraic Topology and Algebraic Geometry that are of such nature.

Against the monopoly of Set Theory and First Order Logic, I attached an appendix of A.M. Vinograndov. We shall continuo our discussion!

Dear Costas,

I've been reading through Edward Nelson's seminal paper on Completed versus Incomplete Infinity in Arithmetic. I must say, it is quite instructive. It strongly reminds me of non-standard models of the "set" N of natural numbers. In this view, Nelson's counting numbers agree with the standard numbers. The next step is to have a corresponding real number system. In non-standard models, it will contain infinitesimals and infinitely large numbers.

The same is probably true for Nelson's approach via Aristotelian numbers. If so, is this a fair price to pay for avoiding the actual infinity of the set N?

Can anyone express self-justificationly what infinity is ?

If no one knows “what infinity is”, many topics relating to “infinity” in RG leads us nowhere.

I sincerely hope that RG can help to solve this problem.

@Geng

A question 'What is infinity' needs a context to be meaningful. The question 'What is an infinite set' has an easy answer: a set X that allows a function X --> X which is injective and not surjective.

If the universe is self-similar, infinity reduces to a finite, 1. 

Dear Ulrich and Cj,

The problem we frequently met in front of our philosophy students and mathematics students really is: “what infinity is and, its properties, relationship with zero etc?”

Dear Cj Nev,

Can you elaborate on your statement "If the universe is self-similar, infinity reduces to a finite, 1."?

Thank you!

@Cj,

Mathematics does not depend on properties of the universe. The fact that mathematics is inspired by reality doesn't make it into a copy of reality.

@Geng

Of course students may be not always aware that general questions laking a specified context may have no reasonable answer. You better tell them this instead of trying to invent some nonsense to answer nonsensical questions. It was Heisenberg, who stressed that our language allows us to form more non-sensical questions than reasonable ones and that even scientists may get trapped by this.

Marcel, If "mathematics does not depend on properties of the universe," on what then does mathematics depend?  Is reality not mathematical?  Certainly any error or incompleteness in mathematics "doesn't make it [mathematics] a copy of reality," however, that is only because reality will make that correction.  If on the other hand you are saying that the universe, or reality, does not depend on mathematics, I would be in full agreement.

Dear Costas, Happy to elaborate.  As we know, that which is self-similar to infinity is of course not the finite, but the infinitesimal, which, as any finite entity does (including the universe), continuously begins and ends (laws of conservation/Noether symmetry, thermodynamics, Bekenstein bound (courtesy of H. Chris Ranford)) through the reality of curved (v. irreality of linear) space (more fully elaborated in 2015 publication, please pardon any vagueness at this time).

@Ulrich,

A function f: X --> X is injective or 1-to-1, provided f(x) = f(y) implies that x = y.

Then consider A = {3, 5, 8}.   For this example, in what sense is a function g: A--> A that is injective but not surjective, infinite?

@James.

If you find a function {3,5,8} -->{3,5,8} which is injective and not surjective you have proven that the set {3,5,8} is infinite. The finiteness or infiniteness of the function is not under consideration. For the set {1,2,3,...} such a proof works by means of the function n |--> 2 n.

@Geng

The challenge for you is to revise your naive understanding of 'general notions' ('universalia' in medieval european philosophy). The notion of 'infinity' has its right to exist as a comprehension of many experiences, observations, and considerations referring to many different contexts. To know 'what infinity is' has no other meaning than having the ability and knowledge to hold enough of this stuff in ones brain to allow to unfold its aspects and interrelations and consequences. Obviously your categories of 'properties' or 'relationship with zero' can only be discussed meaningfully for subsets of the whole stuff. For instance, a 'relation with zero' plays not any role in the context of set cardinalities mentioned by me earlier. It plays a role, however, in the context of arithmetics of numbers ('1/0 = infinity'). 'Knowing what infinity is' as a single insight is an illusion. Learning that illusions do not justify 'live long regret' is important for students. Teaching this by providing interesting and clear examples is a rewarding task for professors. 

@CJ:

If mathematics would depend on physical facts (as perceived by us), the very idea of mathematics being applied to physical problems would be corrupted by circularity.

That is why mathematics must be  an independent "reality" (of axioms and logic). Is the universe a 4D sphere? A torus? A projective space? a  4D Klein Bottle? You name it, we have it. Mathematics is not forced into something, whatever the physical problem at hand may be.

Evidently, axioms are ultimately a human choice (and, to a minor extent, also logic). Yet some of these axioms neglect the highly probable finiteness of the universe. Also, logic implication deviates essentially from physical "cause-and-effect" (see my posts in the thread on mathematics being innate to nature or a human contrivance).

Evidently, mathematics lives only through the acts of mathematicians whose intuitions and motives come from reality; properly, from human reality, which is perceived reality with a rich mental component.

As to your question "Is reality not mathematical?", my answer is:

Mathematics is (at least) helpful to understand reality, and it is so  because mathematics is designed to be independent of reality. Its help would be worthless otherwise.

Dear Ulrich,

thank you！

External users of mathematics may find it weird that its design is said to be independent of physical reality and that this is an indispensable requirement. There is a nice historical illustration of this.

The ancient Greek were convinced that the rational numbers were the only "real" ones, occurring as a result of measurement. It is a quite reasonable belief as simple experiments with a rope, marked every unit length, (seemingly) confirm:

Measure from A to B, then fold the remainder of the rope back to A when B falls within two successive unit marks. Repeat the process when you are back at A, and so on, until you obtain full units.The exact distance AB is the fraction

        (total number of units) / (amount of folding +1)

(the "+1" counts for the successful end). You can do the experiment with great patience, or use the knowledge that the rational numbers are dense within (our) reals to understand why the physical experiment seems affirmative in the sense that the folding process will eventually halt at full units.

Pythagoras' theorem shows that the hypotenuse of a triangle with both rectangular sides of length 1 must have length square root of 2, which turns out to be an irrational number. To put it boldly: the ancient Greeks overlooked an entire range of "algebraic numbers" as possible values. This problem was solved later by extending the real number system to what it is now. This system seems to be better supported by nature (famous transcendental numbers like e and pi are properly included).

The question may arise whether this problem could occur again in our times. Can we be sure that we do not overlook some other range of possible "reals"? The answer isn't a simple "no", but "highly improbable".

It is known today that "our reals" constitute a so-called complete totally ordered field, and that such a structure is unique in the mathematical universe (which contains lots of "competing number systems": computation modulo a prime, p-adic numbers, reals with infinitesimals and infinite numbers, etc.). If we overlooked some range of numbers relevant for physics, the extension is either not a field (the familiar algebraic computation rules break down), or it is not ordered (comparing magnitudes fails partially (complex numbers) or entirely (computation modulo a prime)), or it is not complete (limits, derivatives, and integrals are compromised, e and pi become questionable). If any of these phenomena would show up in reality, our physics would probably be dramatically different.

This illustrates how mathematics can provide confidence in our knowledge by being based on some broad and independent reality of axioms and logic.

>…. some number depending on n, and as n gets larger it gets closer and closer to log(2)….…numbers like e and pi are properly included…

Marcel and All, As mentioned before, I do not confuse the universe with "physical facts," the "physical world," or with mathematics alone, as I have also clarified earlier from the language James Peters provided:  mathematics along with the physics is 1:1 with the universe injectively not surjectively (in other words p, math or physics, implies q, the universe, but q, the universe, does not imply or "depend on" p), while at the same time mathematics along with the physics maintain between themselves a 1:1 relationship both injectively and surjectively. 

In addition, if the commonly-assumed, unsupported claim is made that the universe or reality in which we are inextricably a part and, therefore to which we are inextricably self-similar, is a mere "perception," with what circular perception is such an unperceptive claim made?  What in reality makes those perceptions?

Marcel, in disagreement my response above addresses the first part of your post, which, however, appears to be in agreement with the second part of your post, except for these premises of yours:  "some of these axioms neglect the highly probable finiteness of the universe. Also, logic implication deviates essentially from physical "cause-and-effect."  "Probable" is not actual and "'cause-and-effect'" is not physical.

Marcel, on your latest post above, how could the ancient Greeks have overlooked the irrationals, Pythagoras, an ancient Greek himself, discovered?

@Cj: ...mathematics along with the physics maintain between themselves a 1:1 relationship both injectively and surjectively.

A visualization of injective and byjective maps is shown the attached images, explained in

http://mathworld.wolfram.com/Injection.html

If f:X --> Y is injective, then f(x) = f(y) implies x = y for x,y in X.   Let X be a set of mathematics results and let Y be a set of Physics results.   I agree that that in that case, there is an injective mapping on X into Y.   However, I have difficulty with the second part of what you wrote, namely, an injective mapping on X into Y is also onto (surjective).   For example, Newton's third law (every action has equal and opposite reaction) is a result in physics, arrived at by observation (without mathematics).   Do you agree?    Perhaps you can think of other examples (in mathematics or in physics), where, for example, a result in mathematics has not counterpart in physics or a result in physics that has not counterpart in mathematics.

James, thank you for the further opportunity at clarification.  In your "What is mathematics" question, I believe, I posited that mathematics abstracts from the physical 1to1, bijection; for every mathematical abstraction, there is an equal and non-abstract physical phenomenon.  Advancing your apparent counterexample to this position, Newton's third law -- for every reaction there is an equal and opposite reaction -- note all of the 1to1 corresponding mathematically-abstract terminology:  for:division; every:quantity; reaction:function; there is:existence; an:one; equal:symmetry; and:addition; opposite:negative.  I can provide many more examples for all the laws of physics and for all of mathematical abstractions, but unless taken out of this bijective 1to1 context (i.e., a certain law of physics corresponds to a mathematical abstraction, but not to another and vice versa), no counterfactual examples would be forthcoming.  This stance points as well to the need for a yet-to-be discovered (not invented, like set theory, etc.) concrete, tangible, physical correspondence to the foundation of mathematics in the necessarily self-similar universe.

Cj: (quote) [...] how could the ancient Greeks have overlooked the irrationals, Pythagoras, an ancient Greek himself, discovered? (unquote)

It is said that a member of Pythagoras' community found the argument that square root of 2 is not a fraction. As the legend goes, the man was banned from the community and  a symbolic grave was dug for him. This may give an idea of how the discovery was digested by the Greec civilization of that time. They only discovered that their nice number system didn't fit, but they had no clear answer to it.

@CJ: (quote) 'cause-and-effect' is not physical (unquote).

I am referring to simple reality. It you step out of a window at the first floor (cause), you will fall down 3 meters (effect). If you touch a hot plate (cause) your hand gets burned (effect). Things that happen cause other things to happen. This (naive?) kind of "causality" is often confused with the meaning of logic implication, to which it resembles only partially.

@CJ: (quote) "Probable" is not actual (unquote).

Though widely believed, there is no evidence for- or general agreement on- finiteness of the universe. Hence "probable finiteness" is appropriate. My point is that the standard axioms of set theory (ZFC) hold the mathematical universe to be infinite.

@Cj: (quote) [...] mathematics along with the physics maintain between themselves a 1:1 relationship both injectively and surjectively. (unquote).

In addition to James' remarks on this quote, note that mathematical concepts apply strictly to mathematical objects. All other uses should be seen as imaginative language or poetic freedom. Even with this annotation, the quote's content oversimplifies the complex relationship of mathematics with physics.

>...the man was banned from the community and  a symbolic grave was dug for him...

Panagiotis,

You may be right. The community around Pythagoras was a kind of religious sect with strong interests in numerology and its members may have been submitted to an oath of secrecy.

The story as I heard it explained the furious reaction on the discovery of an irrational number as mere anger for damaging (destroying?) a picture of a "Platonic world". After all, Pythagoras' theorem also became public (it might have been discovered before Pythagoras). 

Marcel, simplifies, yes; oversimplifies, entirely doubtful.

CJ, there is no such thing as a 1-1 correspondence between two sciences, not even approximately, especially not if mathematics is one of them. Do you realize that the content of mathematics can be quite different if assumptions are changed? Replace standard logic by intuitionism, and (among other things) calculus drops down. Change the axiom system ZFC into AST (altenative set theory) and all infinite sets are gone. That ZFC + usual first order predicate logic are more or less the standard is partially due to the internal dynamics of research and publishing activities. Alternatives have their reasons and none can be seen as a self-evident choice or nature's choice.

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, meeting the challenge?

Dear Marcel, thank you.

Remember reading[ about PyTHAGORAS CONCEPT OF THE WORLD] that everything is being upon WEHOLE NUMBERS RAISED continually to POWERS [ which, according to my understanding involves  also ROOTS of.Work of mine in the following link presents solutions of Classical Problems, according l concepts of my Logic:             http://www.stefanides.gr/Html/Classical_Problems_et_Alii_with_Web_Links.htm

The Quadrature by Ruler and Compass is shown if the following Links:

http://www.stefanides.gr/Html/piquad.htm

http://www.stefanides.gr/Html/QuadCirc.htm

The Photo, underneath is a very good approximation of the Doubling of the Cube Graphically by Ruler and Compass         

It gives a value of T^3 = 2.0,... [ http://www.stefanides.gr/Html/Nautilus.htm ].

Regards

James, Can I understand why you would upvote Marcel's last post?  It comes across as pure condescending dogma to me, that's several times now.

Geng,

reality does not respect the notions that humans formed in the past. Small quantum mechanical systems are neither waves nor particles; why mathematical quantities should not behave like numbers and not like numbers? We better adjust our notions to what we observe in nature and in consistent mental edifices (theories).

CJ, if the way I express things has offended you, I do apologize for this. I assure you that this is not intentional.

On the other hand, this is a discussion forum and participants are not expected to agree on every issue.

I expressed that mathematics has various faces. The face of ZFC + standard logic is very supportive to theoretical and applied physics. Other faces are (to put it carefully) not yet known to be suited for the job. None of them presents itself as the most natural one (e.g., ZFC assumes the axiom of choice, which in the past has been debated heavily with qualifications ranging between "too trivial to be stated" and "too strong to be true"). 

So, the statement about a 1-1 correspondence (or something close to it) between math and physics needs more specific arguments.

Marcel, I appreciate and accept your explanation and understand your focus on past up to presentday schools of thought, with which scholars would be as familiar, however, as familiar that they are also not the end all.  My focus is to advance those incomplete, sometimes logically fallacious stances with a new and unique proposal, I have posted many times, is under publication due out in 2015, which naturally addresses in complete and specific detail existing arguments on mathematics along with presenting and leading to the aforementioned new proposal that does link mathematics with physics 11, etc., beginning with a non-circular foundation of mathematics.  You would have to admit that even at least one definition of the abstracted (from the physical) mathematical concept of infinity is physical (e.g., curved spacetime), hence 11?!

So I do apologize I cannot provide the specifics at this time and will try to be sensitive to that shortcoming and avoid being too infomercial-y until 2015.  Truthfully, I am testing the validity of this proposal among the most thoughtful researchers any of us could come across here. Thank you, again, and kind regards.

I would like to comment on the correspondence of theories. Usually this correspondence is not a set-theoretic map, and not necessary 1-1. It is rather a functor between categories. In model theory there is "interpretations", see e.g W. Hodges, Model Theory, Chapter 5. 

I will attached Rosetta Stone for categries, which give another correspondence:

Thanks, Costas, that article provides a lot of one-to-one correspondence examples between physics and topology (mathematics) James asked for earlier, perhaps in the what is mathematics post.  If I may ask under this post, James and All, are there any specific counterexamples to 1to1 correspondences between physics and mathematics?

Dear Cj,

Do not rash! When we have an analogy, even in axiom systems, this does not mean that the whole theories are similar. This correspondences work rather as "metaphors" which might lead to new theorems. The important thing is that the connection it is not always i-i, and it is rather structure preserving maps, ie homomorphisms, functors etc. 

As for the counterexample, I may use a "square" in mathematics and this is unique, whereas in physics there are many material squares! Try to see the general picture and free yourself from this 1-1 mappings. 

Dear Panagiotis: 

I took a look at your website

http://www.stefanides.gr/Html/QuadCirc.htm

where, to my surprise, you seem to find that pi is a root of the polynomial x^4+16x^2-16.

A famous theorem of Lindemann (1882) states that pi is transcendental, meaning that it is never a root of a rational polynomial. This is considered one of the main achievements of 19th century mathematics!

I checked that (pi)^4+16(pi)^2-16 is about −0.677238549 using pi = 3.14159265359.

The value of pi derived from your root is 3.144605511.., which is somewhat worse than the oldest known approximation 22/7 = 3.142857143.. of pi...N.B.:  I used a screen calculator for my computations (I am currently unable to get into Mathematica).

To be helpful, I noticed a wrong value for tan(theta), given that sin(theta) = pi/4. Just check this with tan(theta) = sin(theta)/cos(theta) replacing cos with sqrt(1-sin^2).

I hope you are aware of the problems and desillusions awaiting people claiming to have solved the quadrature of the circle. It is the perpetuum mobile of mathematics....

Wish you good luck and much wisdom.

@CJ and Costas,

Talking about models...

Traditional first order predicate logic also allows to translate or to interpret one logic theory (axiom system) into another one by a language transformation. These terms have a technical description the meaning of which, however, is close to what one should expect of it. The language transformation resembles an injective transformation, but the most relevant property it can have, is to be faithful or conservative. This means that a translated or interpreted formula can be proved in the new theory if and only if the original formula can be proved in the original theory. 

This is strictly a matter of formal logic. I've never seen something like this being used in a different environment.

>..where, to my surprise, you seem to find that pi is a root of the polynomial x^4+16x^2-16...

Costas, no problem.

Dear Panagiotis,

"equation is x^4+16x^2- 16^2=0"

I made a typing error at the constant. In my notes I found

3.14159265359^4+16×3.14159265359^2−4^4 = −0.677238549

which uses the correct polynomial with the same result. Given that I use a value of pi that is correct up to 10^{-11}, the computation should be correct up to 10^{-9}.

I see that you are quite active. Your second link contains different equalities of type "pi =value", unfortunately with Greec comments which I do not understand. I hope you are not disputing the known value of pi...

Dear Marcel,

My theory is based on a Specia Ortogonal  Triangle : "The Quadrature Triangle,"

as defined it, since its horizontal [ smaller] side  multiplied by its hypotenuse is the square of the other side [ vertical- bigger].

[ having a unique angle  : ArcTan( T), where T is the Square Root of the Golden Number.

This, related to circle, gives the relevant relationships.

Derived it by analyzing sections of the Platonic Timaeus.

[Paper Presentation and Proceedings Publication,

 “Golden Root Symmetries of Geometric Forms” The journal of the                

 Symmetry: Culture and Science,Volume 17,Numbers 1-2, 2006, pp 97-111

  Editor: Gyorgy  Darvas.

  Conference :SYMMETRY FESTIVAL 2006,  BUDAPEST HUNGARY- 

http://www.stefanides.gr/Html/GOLDEN_ROOT_SYMMETRIES.htm ].

A similar to this[ but not the same] is the Kepler/magirus  triangle, known for over 400 years but not searched its qualities  extensively [ as far as I know],  Ref Kepler/(Magirus):

[KEPLER TRIANGLE given to him by a Certain MAGIRUS, according to the book “ A Mathematical History of Division in Extreme and Mean Ratio by Prof.Roger Herz-Fischler[ Wilfrid Laurier University Press 1987].

Indeer current transcendental pi value[ 3.141592654…] does not satisfy quadrature :

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_11.pdf

[ Attached photos below] 

But he ittationl  value  4/sqrt(Phi) does :

http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf

 With reference to above

-----------------------------------

ATTACH BOOK Concerning method deriving above mentioned equation:

 π^4 + 4^2*π^2 – 4^4 = 0

Relations  of a square [4x4]  and its formation into a cylinderof circumference 4 and height 4.

Ref pages 43-47 of book, in link:

http://www.stefanides.gr/pdf/BOOK_1997.pdf

Dear Ulrich,

Thank you for the very good example of “neither waves nor particles” in physics about small mathematical quantities of variables, but the problem is no one can tell theoretically and practically in mathematics so far “when do they behave like numbers and when not like numbers?” -------- that is why we have suspended “infinitude related paradoxes families”.

I appreciate very much  your direct and frank opinion.

Regards

@Panagiotis

The 'equation'  π^4 + 4^2*π^2 – 4^4 = 0

says

π < 3

I can't believe that any ancient mathematician in his right mind fell behind the trivial

insight that π > 3.