Today I'd found in Frankfurter Allgemeine Zeitung a review of a book by Klaus Mainzer "Die Berechnung der Welt. Von der Weltformel zu Big Data." (At present only available in German). Following the revue, the message is an attack against Chris Anderson, former CEO of the journal "Wired", who wrote 6 years ago the book "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete". Good punch, good salesman.

In the a.m. review by Helmut Mayer about the counterstrike by Mainzer, the reviewer declares (among other relevant points) that data patterns do nothing explain. Right!

Here we just are at our point of discussion. A critical point of any finite experiment (in nature or on the computer) is extrapolation of the result. Extrapolation is a matter of theory, and the kernel of theory is abstraction, mathematically spoken: infinity or its reciprocal value. Theory rests on elementary patterns in our brain, e.g. concepts like point mass, point charge, trajectory or line, abstract objects which are infinitely thin or small etc. Thats the core of Theoretical Physics and helps to extrapolate experiences. Well, you can extrapolate with a computer, but then you have beforenad programmed something, a linear relation etc. You dont get rid of the need for theory. The "concept of infinity" forms its kernel, which acts like the kernel of a living cell: without it, life is over. Maybe that some branches of math can survive without this concept. Physics would not.

For mathematics you may be right, but for physics I have to disagree, because some problems can only be solved for boundary conditions and often the value of the boundary condition is only determined in infinite distance. A simple example would be a gravitational or electrical field in vacuum - the boundary condition is than that at x = inf the value of the field = 0.

Another example from statistical physics is a distribuition function which gives you the thermodynamical data (i. e. particle density for example) as statistical moment. This can only be done by integration over the whole (velocity) space. Hence you have to use improper integrals with infinity as integral boundary.

I know of no other concept which would allow to gain the same entities without using the concept of infinity.

@Johannes. In 'finite physics' you have, of course, a finite space for your fields and a finite velocity range for your statistical mechanics example. If this looks strange to you, observe that each computational model (also one which uses 10000000000000000000000000000000000000000000000000000000000000000 bits for the representation of a real number) of the phenomena under consideration has just this situation. You probably don't doubt that such models can describe their phenomena in agreement with empirical facts (where agreement does not mean mathematical equality, since empirical data can never be obtained with the accuracy inherent in such computational models).

@Simen. Perfectly d'accord. Of course, my topic here is exclusively what you, doing justice to the historical development, call actual infinity. What I want to stress: Any mathematical problem that can be carried over to the computer does without actual infinity. Fractals, for instance, can very well carried over to the computer. Anything what is interesting about them can be shown by finite (but extensible !) series of finite pixel graphics.

Do you mean actual or potential infinity?

@Juan-Esteban, the axiom of infinity say that there exists an infinite set, so in my opinion it is "actual infinity". You don't need an axiom to consider potential infinity : if you need a set greater than the finite one you consider you can construct a bigger one, the axiom is not needed to do that.

I may add that even to considering a real number you need the axiom of infinity. The whole analysis is based on it and from an heuristic view point it is really useful (even not necessarily from the computational one).

The concept of infinity in both mathematics and physics is a philosophical idealization which make "life" in lots of aspects easier (solving ODEs is "easier" than solving difference equations, certain functions vanish at infinity, etc.). However, if we want to exclude "infinity" from our live in a consequent manner, we have to set up a theory whose mathematics is based entirely on e.g. finite fields (Galois fields). After all, fields are mathematically speaking nicer than rings or groups only. And here we don't need to worry about Goedel's theorem at all (proved in a paper by Ax in 1960 before he opened a hedge fund on Wallstreet). It only gets tricky when we start looking at extension fields of our original Galois field so that we can solve all possible (finitely many) polynomial equations of finite degree (up to the characteristics of the field minus 1). This chain of extensions is unfortunately infinite - in contrast to our complex number system by Gauss, in which any polynomial equation is eventually solvable.

Now coming back to physics, there have been attempts in the 1950s by some Finish physicists and astronomers to formulate a physics entirely based on finite fields (Järnefeldt and Kustaanheimo in Helsiki). However, the interest of the scientific community has died out with them. Later on p-adic physics tried to capture some of the aspects as to handle "infinities" and "infinitesimal" quantities in an easier fashion but like the Galois fields they p-adic numbers are not closed, i.e. there are infinite algebraic extensions again.

Interestingly enough, for certain large primes p the Galois field GF(p) looks locally around 0 like a "large" subset of the positive and negative integers (including a well-ordering). So for sufficiently large and suitable primes p and using the Planck scale as the minimum unit for counting/measuring physical observables (distances, mass, time,...) we might not really need anything else in physics...

@Juan-Esteban Palomar Tarancon:

In the note directly preceeding yours I wrote explicitely that I speak about actual infinity.

@Leo.

I fully agree with your first paragraph.

Your second paragraph can easily be misunderstood. If you cut any real number to a fixed number of representable bits, you loose nothing from the point of view of application to real word problems , if this number of bits is large enough. Then the representable rational numbers are so dense in the hypothetical continuum R that they allow to 'simulate R' sufficiently accurately for all practical purposes. Defining real numbers as equivalence classes of Cauchy sequences of rational numbers (as it was done in the first academic analysis course I attended in 1962) is very close to this point of view. If one treats the available number of bits as variable, no relevant difference between these viewpoints remains.

Ulrich you said: "Anything that is interesting about them can be shown by finite (but extensible !) series of finite pixel graphics."

Perhaps here by "interesting" you mean "calculable".

There are things that are interesting but not calculable. For example can you guarantee that the physical constants (natural units just carry the problem elsewhere) are rationals?

Moreover if you rely only on pixel graphics, then a strange problem arises: Geometric shape definitions would depend on imaging technology, e.g. the definition of a line would be provisional until the next update of LCD or plasma or paper quality. But exactly the opposite is true: We have a platonic perfect square with sqrt2 diagonal and any technology strives towards that standard.

Yes we do

@Chris. Are you speaking ex cathedra? Normally only popes do so.

@Ioannis. I should be more explicit on my picture of finite physics: I don't rely on physical constants being rational (which in my eyes would be a meaningless assumption). Instead I assume, that a finite and technically providable number of bits for the representation for reals ( e.g. 32 bit for float, 64 bit for double, 80 bit for long double, and up to several 1000 bit for type mp::real on my C++ system) allows a sufficiently accurate coding of the values of physical constants that no conflicts with empirical facts result from the approximation error. At present this accuracy is given for all physical constants certainly with reals of type double. To your final argument: I used 'pixel graphics' as the name of a data structure not for the light distribution generated by some display device (which would include a copy of 'The Beauty of Fractals').

@Markus. Since a few years the official ehemerides of the solar system are made by solving difference equations and not by ODE's (i.e. they are obtained by numerical integration and not no longer by improved forms of Newcomb's and Brown's series expansions). Where accuracy is a must, numerical methods are mandatory (assuming 'real world' systems and accuracy defined as agreement with empirical facts). The Galois field approach suggests itself to algebraists. From my point of view it is a very unnatural idealization. A more natural one is that provided by the algorithms of numerical mathematics: numbers are finite bit strings and the arithmetic operations satisfy the field axioms not exactly but 'up to numerical noise' (which for a state of the art implementation is minimal in a sense which in most books is formulated in a cumbersome way that looks not attractive to a mathematician).

@Simen. You can't see the point. My point is one of unification: Also the activity of modelling (simulating) physical (or engineering, or biological) systems has a home in pure mathematics, the theory of hereditarily finite sets. Having learned Euler-style analysis from old books as a school boy, changed then to the comfortably closed world-view of the Bourbaki-school during university education, and finally having learned to simulate (by C++ programs) physical processes in machines during R&D work in industry, I collected experience with so many modes of looking to physical systems that a desire for unification grew. I did not expect that the solution would be so simple. Is it more than simply a bit of common sense? Yes, where is the point?

So, if I got it right, you are saying that you actually don't care if infinite sets exist or not, but you can always do your job by relying only on finite representations of physical and mathematical entities. And then, you give in to the temptation to *define* reality as exactly those things that are finitely representable. Right?

@Simen. Expressed more carefully, my intention is not to throw away the axiom of infinity in mathematics, but to find out for which parts of mathematics it is needed and for which parts not. Think of the corresponding question concerning the axiom of choice as an analog. Also here, the most reasonable position is not to throw it away, but to clarify which theorems can be proved by accepting it (e.g. every linear space has a Hamel basis) and which theorems can be proved by not accepting it (e.g. every function R->R is measurable). For the mathematics of hereditarily finite sets the strong point is that it covers all computational physics. BTW your hint concerning necessary infinity in finite math is a misunderstanding, in my mind. If you would indicate why the reference you gave should support your claim I'm ready to try convincing you of the contrary.

@Simen.

I think presently there is enough 'messing with the foundations of mathematics' going on from the side of mathematicians that applied scientists should find what they need if they are willing to dig. For instance, a constructive definition of natural numbers N (and Z, Q), is well established, although the much inferior predicative axiomatic definitions, e.g. Peano arithmetic, or Peano axioms (requiring second order logic) are more widely known and and applied. More general, constructive mathematics combined with the syntactic inovations which emerged from Object Oriented Programming could provide an adequate framework for natural sciences.

I found your remark concerning spectra of quantum mechanical observables interesting. Approached from a natural point of view, the problem does not exist: Of course, if the configuration space (on which the wave functions are complex-valued functions) is a finite lattice of points, the Hilbert space of such wavefunctions is finite-dimensional and any symmetric operator is self-adjoint and the spectrum of such an operator is discrete. What in the infinite dimensional case is a continuous spectrum, in our finite-dimensional approximation is a discrete spectrum with closely packed points. The spacing of spectral points may vary over many orders of magnitude and may change with the position. The operator thus may show a very interesting 'spectral behavior'. An example for 'finite quantum mechanics' is given in the work in the following link.

http://www.ma.utexas.edu/mp_arc/c/06/06-356.pdf

More on

Hereditarily Finite Sets

can be found in the attached file.

I don't succeed in attaching more than one file in one post, so I have to spend two more posts (despite Occams words: 'entia non multiplicanda praeter necessitatem')

See previous post

Dear Ulrich,

You could have applied Occam's razor by transferring all these files to a single folder compressed using winrar. :)

Dear Ulrich,

Let us assume that the concept of infinity had never been there. Shall we then ever be able to complete enumerating the problems that we would have faced in studying mathematics as well as physics?

Dear Hemanta,

as already pointed out in the second contribution (by Simen) it is only 'actual infinity' that would disappear when we would abolish the 'axiom of infinity' in set theory. So called 'potential infinity', which to Carl Friedrich Gauss was the only legitimate form of infinity, would be sufficient to account at least for my problems.

Thinking pragmatically, it can only be encouraged to investigate what remains of mathematics if the axiom of infinity is removed. Whatever the outcome, we will get better informed about the (un)avoidability of the infinite.

It should be noted that there are no acute problems with infinity in mathematics. We are no longer haunted by the old Greec paradoxes about limits, or the early paradoxes with naive set theory. The issue is largely a philosophical one (or a pragmatic one, if you wish) and the mere occurrence of a phenomenon like the Strengthened Finite Ramsey Theorem is actually quite exciting.

An interesting and entertaining view is N J Wildberger's in:

http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf

About Wildberger's manifest.

This paper makes a number of quite strong statements about the content and methodology of mathematics. After five pages, I already collected so many objections that I had to pause to comment on just a few of them. Glimpsing through the rest of the paper, I see there is a lot more to object to.

(page 3, text lines 1-14)

Problems with mathematical education are *not* a symptom of mathematics "not making complete sense". With the availability of internet, i-pads, etc., we are living at a turning point in educational history, and the problem of collapsing old education methods is omnipresent at schools, not only with mathematics.

Moreover, the proposed remedy ("to provide a proper logical framework for those who might have an interest in a scientific or mathematical career") is highly unrealistic:

what about those students who are not the most brilliant among the brilliant ones?

I'm following an internet discussion of the Science Math Primary/Secondary Education Group. If there is one thing they are *not* waiting for, it's to push 18_ students into rigorous definitions of numbers, fractions, reals, polynomials, and whatever.

(page 5, text lines 1-20)

The Russel paradox is basically the following formal statement in first-order predicate logic: (exists x)(all y) (P(y,x) not P(y,y)).

This statement is a plain "contradiction", exactly like statements of type "P and not P".

Do you see the notion of an infinite set in it? Its translation into set theory is: there exists a set containing precisely those sets which are not a member of themselves.

Is this a paradox? No, there just isn't a set with these requirements. It just means that naive set theory cannot be held. An entity (e.g., a round square, a blue object that is green) need not exist just because you are able to describe it.

The only dubious fact about Cantor's original proposal was that it was not organised by "rules of behavior" (i.e., axioms). This has been taken care of later by Zermelo, Frankel, Von Neumann, Bernays, and many others. In this way, set theory is no different from group theory, the theory of partial order, or theory you-name-it. And most students are at comfort with the "naive viewpoint" to deal with elementary sets.

(page 5, textlines 21-30)

Considering the "infinite set of infinite sets" as a self-referential paradox is a wrong suggestion: the "set" of all sets already leads to a paradox in naive set theory due to Cantors result that the "power set of a set" has a larger magnitude that the original set, not because it is self-referential. Axiomatic set theory so far produced no paradoxes and that's the best thing that can be said about *any* mathematical theory.

Wildberger's style is very convincing to the outsider, but his judgement is often unbalanced. For one more example, nfinite sets are *not defined* as "sets that are not finite", in fact, it is quite the opposite: as mathematicians often attach new or deviating meanings to ordinary words (e.g., the adjective "normal"), they may feel a need to assure that a term like "infinite" is used in its ordinary meaning of "non-finite".

If this paper needs to convince people to restrict mathematics to the finite case,

it could have done which more moderate and balanced views. Just make a proposal (e.g., leaving out the axiom of infinity), and see what you can do with it. There's nothing wrong with (modern) mathematics, but if one knows a way to make it better, please go ahead.

@Marcel,

Wildberger categorizes these pages as a 'view', and he certainly does not avoid being blatant. I looked a bit in his scientific papers and concluded that he knows how to do traditional mathematics.

His main message that 'it makes not complete sense' to build the main body of mathematics on believes (e.g. non-contradictory nature of ZFC) and to appeal to infinite sets (in form of equivalence classes) when trying to give exact meaning to such elementary things as fractions (to build the rational numbers) looks sound to me and supports the idea that there is, in fact, something wrong with modern mathematics:

If I look arround in the RG-discussions, how often Gödel's theorems are invoked as support for absurde claims, and when I remember the shock I felt when my math teacher explained me that although all things in mathematics can be proven exactly within mathematics, unfortunately one cannot prove that mathematics itself is internally free of contradictions, I can't but consider it a failure not to make public the fact that all mathematics that is relevant to real world problems is not touched by these foundational difficulties.

And that these difficulties are restricted to those areas of mathematics for which infinite sets are required not only as growing but as completed entities.

I'm quite sure that we will not have to wait longer than till 2050 that the former part of mathematics will be codified in a nice book and a nice computer program under a title like 'Mathematics for the Natural Sciences'. This book will contain all the material of Bronstein/Semendiajew, Abramowitz/Stegun,..and much more with all things proved by the built-in proof machine. It is obvious that much work has to be done till 2050!

Coming back to the main theme, Michael Kiessling gave a talk recently which he summed up thus:

Quote

Michael Kiessling, "Infinite in all directions"

In a recent BBC interview one of my colleagues, ``Dr. Z.'', suggested that there are only finitely many integers.

When the perplexed interviewer pressed him, Dr. Z. answered something like this: `I know this sounds nonsensical, but it makes more sense than the mess you get yourself into when you allow infinitely many integers.' I am not sure I would share his radical sentiment, but infinity is a tricky business, indeed.

The Lorentz force on a point charge can even be infinite in all directions, and this has gotten people into a fine mess. I will explain how physicists computed solutions to electromagnetic equations which don't have any solutions, and how on other occasions they proved the absence of solutions which do exist. Underlying this mess is sometimes a (little too) careless handling of mathematics, but more often a confusion about important conceptual notions in physics.

I conclude with an uncensored inquiry into the life of naked singularities in general relativity.

Unquote

@Ulrich:

Around 1972 the Polish mathematician Andrzej Tribulec introduced the Mizar language to write down formal definitions and proofs. The system includes a proof checker that verifies proofs written in that language. Remarkably, the system uses the Grothendieck-Tarski axioms of set theory, which is properly stronger than ZFC.

Nowadays, there is already a huge data base of verified theorems.

The official website is http://mizar.org/

Just a critical note.

To deal with the set of integers in the range from one to one billion, it is not necessary to run a process listing all the intended numbers. Set theory allows to declare this set as

{ x : x integer and 1

Marcel,

1. I read about the mizar project already, and will take your note as motivation to look closer into it, thanks.

2. I could not agree more. But the pool of problems mentioned in my previous note remains: We have a mathematics that can't be proved to be free of contradictions, and in which we handle objects that exist only by reference to axioms from which we only believe that they are not contradictory.

My rational for refuting infinite sets is not that I consider 'containers with unlimited capacity' a logical monstrosity (being aware that one can interprete sets just as predicates instead of containers). The rational is that negating the axiom of infinity is the simplest (and probably most natural) way to make the remaining mathematics amenable to simle finitist methods as they are used in mathematical logic to handle the constructs of formal languages. Actually, since the universe of hereditarily finite sets can be realized as a formal language with '{' and '}' as the only non-logical symbols all the typical 'mathematical' foundational problems seem no longer to apply.

What kind of foundational problems can be shine up in formal languages is not really clear to me, but I think that someting for which one can build and test compilers is more solid ground than ZFC-axioms.

That totally finite mathematics is suffcient for all real world problems is tried to make plausible in the following essay:

http://www.ulrichmutze.de/rubystuff/rnumrational.txt

Hereditarily finite sets together constitute a model of ZFC without the axiom of infinity. In fact, they are a model of a much smaller portion of ZFC known as "General Set Theory" (GST). I'm not well acquainted with it, but (@Ulrich) it may be worth taking a look at.

Marcel,

thank you very much for this hint. I really benefit from your knowledge of a field in which I am an amateur. The author of the Wikipedia article on "General Set Theory" holds the selfsuggesting (but wrong, as I see it) view that "General Set Theory" is unsuitable as a basis for analysis and geometry.

As carried out in the link to my last contribution, we have only to use the approximations to R that numerical analysis provides, with the number of bits to be used for encoding a real number (typically 64 or 80 these days) being a variable n that is held constant for a whole computational task (for which a typical example is the production of all diagrams and numerical tables of a publication). Then n gets increased in steps till the results of the computational task, within the accuracy required by the task, do not longer depend on n.

A further difference of my appoach to finite sets and "General Set Theory" is that the latter relies on first order logic and axioms, wheras mine relies on direct construction (as formalized in Backus Naur notation) and algorithms that are guaranteed to halt. I would be gratefull to learn whether and where such an approch is in use in mathematics; isn't all of computer science grounded this way?

The author of the Wikipedia article on General Set Theory probably means that GST is insufficient to reproduce analysis and geometry as we know it, which is fairly obvious.

As to numerical computations with a required accuracy, software like Mathematica is very good at computing with arbitrary precision. (It can also handle exact symbolic computations.)

@Ulrich: Your "binary string" modely satisfies all of Zermelo-Frankel set theory except the axiom of infinity, so GST is not recommendable anyway, being too poor.

I don't know how far your ambitions reach. If you aim at a strictly finitary mathematical modeling of physics, some explanation is required both at the physics and the mathematics side. If you just wish to do approximate computations on physical problems, you don't need additional theory or foundations and existing software may probably do the job.

Your last question seems to be one of algorithm correctness. This is an active area of research at the border between mathematics and computer science and is beyond my competence.

Marcel,

how do you substantiate that GST is too poor, where it is obvious that anything that ever will be done on a computer can be formulated and done within GST?

As I pointed out, I don't define hereditarily finite sets by axioms but by construction, using finitist methods that logic uses in handling expressions. This would provide a coherent basis at least for computer science.

I can't see in which sense my question concerning the role of constructive foundations of mathematics is primarily one of algorithm correctness.

Ulrich,

GST is a formal logic system. What you construct is a model of some logic. GST is too weak in the sense that your model satisfies heavier axioms, namely ZF minus the axiom of infinity. Taking your model seriously, you shouldn't neglect the additional axioms (which, after all, are independent of GST and are valid in your model).

(quote) I can't see in which sense my question concerning the role of constructive foundations of mathematics is primarily one of algorithm correctness. (unquote)

There is some (mutual) misunderstanding here.

Doing computations involves more than compiling a syntactically correct program and pressing the "enter" button to put it at work. It may involve modeling a problem (perhaps taken from physics) into mathematical terms with additional reasoning on the available mathematical information, even before you write the first line of a program. You will have a hard time reasoning on a hard-wired model; that's were the logic axioms become handsome.

Apart from this, algorithms going beyond routine computations may need a correctness proof. This is not just about being sure that your programming instructions with "if, then, else, for, while" are doing exactly what you intend them to do. Even at the mathematical level, it is not always clear that an algorithm produces the desired result (e.g., Euclid's algorithm for greatest common divisors, or the construction of a maximal flow in a network). Here, too, (formal) reasoning may be indispensable.

One more remark. Aren't you a bit over-optimistic about computation power? Even problems with a restricted bit-precision can easily run out of hand. My recent research project (submitted for publication) is about design of pseudo-random number generators (64 or 128 bit numbers). Except for quality testing with statistical software, the period must be certified. For the largest design, I'm talking about 2 to the power 44447 (there are pragmatic reasons for making them so big). There is no way to test this. It has been proven with formal algebra using design details. (The proof needs computational support.)

Dear UlrichMutze, First consider mathematics as it applies to physics or the real world. Since Physics has to conform to the empirical world. Further, in mathematics there exist two theorems: (1) a discontinuous function cannot be differentiated, and (2) a discontinuous function can be differentiated. So, in my opinion I strongly beleive that "we need the concept of Infinity in Mathematics and Physics".

Afaq,

if you define

f'(x) := ( f(x+h) - f(x-h) ) / (2h) 'for suitable h'

you only need to define what 'for suitable h' is to mean. If you work on a lattice of numbers instead on a continuum (as explained in the primary post) this has to be done more in the spirit of numerical analysis than in the spirit of limits in R. However you do this in detail, it will cope much better with any challenge coming from the 'empirical world' than the definition of differentiation that we learned in traditional Mathematics.

Dear all, the notion of infinity is even a central element of math and absolutely indispensable! Indirectly it is a formalized expression for "essential" or "non essential" and is well married with any idealization. Math rests on a somehow built-in capability of our brains, like in few other animals, too. Counting and orientation in space are here essentials. Nevertheless, without generalization and notation in an extra-individual language we cannot talk about math.

The problem here is generalization. If we go too far, then we end at "not even wrong" games and cargo science in the sense of Feynman. The fields of cosmology and turbulence theory are filled with this sort. Asymptotics, differential equations -- all are "clients" of infinity. Without this notion whole branches of math would collapse. The math of phase transitions with the need to differentiate discontinuous functions, the calculus of Dirac functions and distribution theory -- all these branches would collapse. I love infinity because it is so useful and for me personally it has been extremely profitable!

I agree that in today's computational sciences infinity stands like a stranger a the corner. However, in theory building, i.e. before we write codes, before we decide which differential equation is needed etc., Infinity is indispensable.

@Helmut

Your contribution would be more interesting if it would avoid unreflected ex cathetra statements about topics already discussed with some care in the present discussion.

@Ulrich, indeed, I have not seen the other contributions as the browser showed only the last few. My mistake, I had to push a button. However, I do not agree with you and you did not answer my comments. Your introductory statements ("Do we need ...? We dont!") are too general. Analysis is completely filtered out. Actually you mean only sets. But math is a much broader ecosystem (and still a human adventure). Wildberger's comment is a fruitful provocation. Like chemistry cannot be boiled down to physics or quantum mechanics, also mathematics cannot be boiled down to GST or logic or both. Your initial question is more of a philosophical and/or science-sociological character, if you allow.

My personal experience with inf is as follows. During the study of turbulence I got a closed system of equations free of empirical parameters. The solutions coincide with observations. In particular I could calculate universal constants of turbulence, e.g. the (dimensionless) von-Karman constant appears as 1/sqrt(2 pi) ~ 0.399 (Its international standard value is 0.4, just recently again reproduced at U. Bundeswehr in Munich with high accuracy). As an aside: Lew Landau still believed that von-Karman can only be measured. I could do this theory only through thinking about singular perturbation and how to avoid all the mess and traps connected with it, and how to make a big cut in turbulence theory by simply forgetting Navier-Stokes and starting right from the beginning with Euler, i.e. with an infinitely large Reynolds number. It was a success with domino effect as it opened also the door to stratified and even non-Newtonian turbulence. The most important part is presented here:

doi:10.1088/0031-8949/2013/T155/014001, next parts come this August at the Abdus Sallam ICTP in Trieste.

@Helmut

The Bourbaki school has demonstrated that all mathematics can be developed out of ZFC(Zermelo,Fraenkel,Choice)-set theory. So your idea that a set based mathematics would exclude analysis is not understandable for me. Your anti-reductionistic attitute towards the relationship between chemistry, physics, and quantum mechanics may well belong to the science-sociological complex. It can hardly be defended on the basis of scientific principles. As you may have seen from the my contributions before, I'm not at all happy with the set theoretical basis of all mathematics since it makes all mathematical truth dependent on the consistency of an infinity system of axioms which, as Gödel showed, cannot be proved without relying on means outside of ZFC. What I suggested is that the part of mathematics that has applications in natural sciences (and thus of course includes analysis) may be characterized, in a way that deviates as little as possible from what mathematicians are familiar with: Leaving ZFC intact with only one exception: eliminating the 'axiom of infinity'. Particularly Marcel gave arguments that this would probably not be enough to escape the Gödel argument.

What you describe as your experience with infinity could certainly (probably less compelling in your mind) be interpreted without speaking about infinity at all.

Good evening, Ulrich,

>Your anti-reductionistic attitute towards the relationship between

>chemistry, physics, and quantum mechanics may well belong to

>the science-sociological complex. It can hardly be defended on

>the basis of scientific principles.

Ha, good punch. It can be very well defended! If you have only one molecule, nothing can happen. If you have two of them they open a qualitatively new potential world as they might collide and react and form compounds you never have dreamed of. Once you have the new compound molecule, you can explain it by QM, but typically not the other way around. That's the rule when you discover things. It's a pity that Dick Feynman or Peter Woid are not here to join this interesting discussion. For Woid see here:

http://www.amazon.co.uk/Not-Even-Wrong-Continuing-Challenge/dp/0224076051

He is a clear anti-Bourbakist in my interpretation, but a serious observer of our present science world which is filled with cargo science sensu Feynman. For me inf will remain one of the greatest achievements of mathematical thinking, e.g. the point mass, the line etc. Have a good night.

Let's try to answer the question by going back to basics.

Here goes:

1- There is demonstrably an infinite number of finite numbers (i.e., numbers with finite digit expansion)

2- There is demonstrably no way to exclude finite numbers from the envelope of possible future, present, or past needs. In other words, it is impossible to demonstrate that a given (ensemble of, or individual) finite number(s) will never be needed

3- We therefore need infinity ( at least in the narrow case outlined above, which is enough to answer the question.)

The way I understand Ulrich, he just wants to have a formally concise piece of mathematics available in which infinity is not available. Technically, it *can* be realized, e.g., by the Zermelo-Frankel set axioms minus infinity.

It takes some acquaintance with formal logic (+ an appropriate language) to accept the idea that a theory may not support a certain concept, whereas the concept can be used in models of the theory (talking the meta-language of a meta-theory).

E.g., a model of ZF minus infinity may contain finite numbers of any size. The axioms do not allow to collect all numbers into an (infinite) set, and yet you may consider infinite models of that theory. It isn't that strange :

In ZF, you can have sets of any (infinite) size, but you cannot have the set that contains all legitimate objects (it would lead to a Russel's paradox). A model of (ZF or ZFC) set theory is necessarily another set, which itself cannot be called by the primary language.

If ZFC is consistent, there exists a *countable* model of it. You can find all of calculus in it. However, in ZFC, the real line is uncountable. Yet this doesn't conflict with the entire model being countable, because the counting is achieved at the meta-level, not at the primary level.

All this should not bother Ulrich if he wishes to work within ZF minus infinity, as he can leave the "meta" thinking to the logicians. Isn't this what most users of mathematics (and even most mathematicians) do? From a philosophical point of view, it may be uncomfortable for Ulrich that the "meta" level of reasoning is haunted by infinity (note that the "physical" objection to it isn't that strong here).

As to Godel's theorem, Ulrich would like this to be out of the way. If ZF is accepted at the meta-level, I think the theorem applies to ZF minus infinity.

One got used to the existence of a fundamental speed limit in the universe. Godel's theorem might represent a fundamental limit on mind and mathematical logic.

Chris, thanks, your No. 1 is already enough in my admittedly naive view as a physics practicioner (+ former math olympionist). I'd defend inf with my blood!

Much more interesting for me personally is the question of asymptotics and natural constants. It has been shown that the von-Karman constant at very high Reynolds number has the asymptotic value 1/sqrt(2*pi) ~ 0.399. This number is two-fold irrational. a) it contains sqrt (2), b) it contains pi. It reflects properties of pure space. Is pure space a piece of nature? Then sqrt (2) and all the rest of space-time, too? I would say yes, while inf is possibly not? Ulrich posed an interesting question.

Marcel, yours is an interesting take on Gödel's.

The exact opposite view can be defended however, to wit: Gödel's demonstrates that any self-consistent system is ultimately open and can only be wholly apprehended from outside itself - which leads to a recurrence relationship that goes on ad infinitum, and which demonstrates a forcible absence of limits.

In other words - any self-consistent mathematical system E is open - thus, Gödel demonstrates that (E+ɛ) exists .

Now build (E+ɛ) and apply Gödel again, and you necessarily wind up with a buildable [(E+ɛ) + γ] , and so forth ad infinitum: Gödel builds infinity through recurrence. Under that interpretation, far from imposing ultimate limits, Gödel imposes infinity.

Now if you map the math onto the physical world itself (OK, this is viewed as controversial in most quarters), then you end up with infinite physical reality as well ....

Marcel, may I remind you that I primarily covers all the stuff used in natural sciences and engineering and gives clearer and more convincing foundations than relying on the consistency of ZFC. Eliminating 'actual infinity' (not potential one, beware) was only the way that suggested itself to me, based on my experience with computational physics.

For all who can read German I add a link to an obsolete book project of mine, which explains the 'science-sociological character' (Helmut's words) of my attitude to the matter.

http://ulrichmutze.de/pedagogic_stuff/mms1a.html

Perhaps to overcome this conundrum, two kinds of infinity are to be introduced: (1) insurmountable linear/discrete/sequential/discontinuous infinity, however overridden by (2) bounded (h to c^2) infinity in physical, iteratively curved spacetime.

Cj, what you propose as the solution to a conundrum, is for me a conundrum itself. Without further elaboration I see no chance to overcome the latter.

Sorry, Ulrich, true my proposal requires further elaboration, under publication at this time. Still I haven't given away the store, my apologies.

I would not go as far as Helmut to defend infinity with my blood. It is just quite intriguing to see that finite beings in a (probably finite) universe come up with infinity in their thinking. Note that religion, with an omnipotent (i.e., infinitely talented and powerful) deity offering eternal life, is also an example of this. It can not be denied that mathematics (and its applications) enjoyed considerable benefits with a notion of infinity. So I am fully open to the question whether infinity can or can not be dispensed with. My attitude towards Ulrich's project is the same as my attitude towards vegetarianism: I tend to follow, but I do like some piece of meat every now and then.

This being aside, Ulrich's project not only wants to replace traditional infinitary mathematics with finitary approximate computations (which can be justified technically), but also wants to found this on solid and convincing mathematical views not involving infinity. ZF minus infinity can be used as a starting point.

As to Godel's theorem, a technical problem is whether its proof still works for ZF minus infinity if at the meta level (where Godel's reasoning takes place) we also assume ZF minus infinity. Personally, I wouldn't mind if the theorem were still valid, as I do not expect absolute certainty or complete solutions from any human enterprise. (The unique thing about current mathematics is, that its body is just as certain as its foundations and first axioms.)

As I already mentioned in my previous post, meta-thinking is haunted with infinity. The annoyance begins with the usual mathematical definition of a (formal) language of first order logic: it requires an infinite sequence of symbols denoting variables. In a finitary version of this, one assumes a (sufficiently large) finite number of variables. This is known in computer science as "logic with finite resources". It was observed by the famous logician Alfred Tarski that a formalized statement of "composition of (binary) relations is associative" can be made, involving just 3 variables, but its (simple) proof uses 4 variables. Tarski showed that it cannot be achieved with less. Nowadays it is known that formal statements of this kind exist for any any integer >= 3. I do not recall any results on how far this discrepancy can go. Are there provable formal statements with a modest number of variables requiring a huge number of variables for a proof? If so, this could become a more serious problem for a finitary approach to formal languages

@Ulrich, my german is just good enough to read the nice story in your reference. For a while I had the impression that you were setting out for some "abacus" with computer assistance, but near the end of the text I saw the appearance of a familiar kind of axiom system, namely Peano axioms for (N, 0, +), where the axiom scheme for induction is still missing. In a setting of first-order logic, this scheme stands for an infinity of individual axioms, one for each formula of the language. I can understand why you won't get into all this, but without induction, the proof capacity is too low.

@Chris, the way I understand your post is that a consistent but unprovable statement can be added to the axiom system, and that this can be repeated ad libitum. However, Godel does not state that such a sequence of extensions exist, rather, he states that even *if* you would extend Peano arithmetic with any sequence of independent statements, this axiom system would not suffice to achieve completeness. Hence Gödel does not build infinity, he uses it (I took "Gödel" with my mouse; how do you put the ":" on the "o"?).

Marcel,

thank you very much for the the effort you take to discuss our topic. I really would like to spend a night with you to discuss all the questions which mathematical logic poses to my somehow stubborn mind.

You certainly know the introduction into mathematical logic by Ebbinghaus, Flum, and Thomas. I enjoyed the careful description of first order logic there, but it is hard for me to understand that experienced scientists accept such a mental edifice which already assumes natural numbers (e.g. as indexes for an unlimited supply of variables) as given, as a foundational basis for the edifice of mathematics. I had a long e-mail discussion with Ebbinghaus on this and similar questions, and he was very reluctent in admitting any point in my questions. The idea that one has first to construct something to which the quantors (e.g. 'for all x', all dreams, or what?) of predicate logic refer befor the whole game makes sense, couldn't be shared by him. As I see it with accepting predicate logic over an unspecified domain for its quantors is the key point which enables the Gödel attack. No empiricly oriented mathematics needs such an unlimited logic! What I have read about type theory indicates that the 'type-guys' take an approach which is much closer to my thinking but I have no idea whether they have a chance to influence the mainstream.

@Ulrich, a first rapid reply on your remark

"(...)that experienced scientists accept such a mental edifice which already assumes natural numbers (e.g. as indexes for an unlimited supply of variables) as given, as a foundational basis for the edifice of mathematics."

The appearance of numbers as indexes for variables is not (meant to be) a big thing. An infinite amount of variables is required, each identified with a symbolic name. Numbers (with the implicit facility of using the base 10 notation) are the cheapest way of achieving this. No other meaning or properties are being attached to the indices. It is only the infinity that one might object to.

Most users of mathematics use logic only to assist or check their reasoning. They use a mild formalism (some abbreviations) and a small repertoire of classical logic rules. One can go another step further and play the "game of formal deduction" (I prefer Gentzen's "natural deduction") to explore an axiom system. But when you start wondering about your game board (formal language, interpretation) , things get more complex. That's the "meta stuff".

Meta-reasoning is ordinary reasoning (on unusual objects, perhaps) and is therefore subject to its own assumptions (usually, ZF; often ZFC). Definitions of "well-formed formula" and "interpretation into a model" require recursion. I don't know of an approach with ZF minus infinity as the basic assumption for all meta-stuff.

As my work on logic has only been occasional (it never was my primary occupation), I'm not that well aware of what is happening outside mainstream logic. To some extent, I do share your surprise about the edifice of formal logic.

If this may comfort you, I am equally surprised about the "edifice" of elementary particles with behaviors that are unusual in a macroscopic world. A quantum physicist (I forgot his name) once reacted on the discovery of another particle (I also forgot which one) with "who asked for this?".

A Reaction on Ulrich's comment

"The idea that one has first to construct something to which the quantors (e.g. 'for all x', all dreams, or what?) of predicate logic refer before the whole game makes sense, couldn't be shared by [Ebbinghaus]".

Quantifiers with reference like "for all x -- real number" or "there exists x -- integer" are, in fact, often used with informal discourse and they do have some educational advantages. In practice, it would add a symbol (like R for the real system, or Z for the integer system) to the formal language to be used in quantifiers. It can *not* be used as a subject of statements (then you make the implicitly intended "universe of discourse" into a member of itself. The problem is painfully clear with quantifying sets: using the universe of all sets as a potential subject of statements leads at once to Russel's or Cantor's paradox.

I'm not surprised that Ebbinghaus does not agree to use "semantic quantifiers" in formal logic. I guess that most logicians won't. It is seen as one of logic's main achievements to separate language and semantics (models). One can use some part of reality for a conviction that a certain topic is not "floating in the air" (perhaps you mean this kind of semantics). But one cannot use reality as the true semantics of formal logic theories (this would create a circularity problem when using mathematics to explain real phenomena). Formal logic requires mathematical (set-theoretic) models for semantics. As expected, most consistency results (i.e., existence of models) are relative to the consistency of something else. No human activity leads to something that is absolute.

Today I'd found in Frankfurter Allgemeine Zeitung a review of a book by Klaus Mainzer "Die Berechnung der Welt. Von der Weltformel zu Big Data." (At present only available in German). Following the revue, the message is an attack against Chris Anderson, former CEO of the journal "Wired", who wrote 6 years ago the book "The End of Theory: The Data Deluge Makes the Scientific Method Obsolete". Good punch, good salesman.

In the a.m. review by Helmut Mayer about the counterstrike by Mainzer, the reviewer declares (among other relevant points) that data patterns do nothing explain. Right!

Here we just are at our point of discussion. A critical point of any finite experiment (in nature or on the computer) is extrapolation of the result. Extrapolation is a matter of theory, and the kernel of theory is abstraction, mathematically spoken: infinity or its reciprocal value. Theory rests on elementary patterns in our brain, e.g. concepts like point mass, point charge, trajectory or line, abstract objects which are infinitely thin or small etc. Thats the core of Theoretical Physics and helps to extrapolate experiences. Well, you can extrapolate with a computer, but then you have beforenad programmed something, a linear relation etc. You dont get rid of the need for theory. The "concept of infinity" forms its kernel, which acts like the kernel of a living cell: without it, life is over. Maybe that some branches of math can survive without this concept. Physics would not.

Helmut, interesting points, and although I agree with you I can still see the other side's arguments.

David Hilbert once said : "We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual processes of things? In short, can thought be so far removed from reality? Rather is it not clear that, when we think that we have encountered the infinite in some real sense, we have merely been seduced into thinking so by the fact that we often encounter extremely large and extremely small dimensions in reality?"

He went on to remark, "From time immemorial, the infinite has stirred men's emotions more than any other question. Hardly any other idea has stimulated the mind so fruitfully. Yet, no other concept needs clarification more than it does."

To say that infinity exists in manifest, as opposed to potential, reality, you'd first have to define reality - the broadest acceptable definition would be that something exists, whether concrete or abstract, if it has ever existed, or exists, or shall ever exist in any dimension (if other dimensions exist) in all of the universes that have ever existed, exist, or shall ever exist. That include not only material things but all the thoughts ever held by any living entity that has ever lived, live, and will ever live .

But this broad definition does not prove manifest infinity: for instance, whereas it is easy to conceive of infinity, because all we have to do is to think of an endless series of numbers, such as the never-ending series of whole numbers 1, 2,3, 4, 5 .... or, say, of all the possible decimal numbers between 1 and 2, it is equally plain that most of these numbers are not actualized within our thoughts : for example, although we are well aware that the following number :

1.00077365442673992652467899993267635625809881625641552537013

belongs to the infinite series of decimal numbers between 1 and 2, it was not embedded in reality, under any definition of what reality is, before someone thought of that number individually, and thereby at some point in time imported it into the realm of actually existing thoughts (and thereby actualized it.) Now that we have ushered it into awareness, we may go ahead and forget it : it popped up on the radar screen of reality through our thought at some point in time, and now shall remain for ever actualized, having appeared at some point within reality. Before, it had never.

This also leaves vast - in fact, infinite - swathes of potential reality that have never been embedded within actual reality, whether abstract or material.

Then there are the different levels of infinity: aleph-1 does demonstrably exist if space is continuous, for instance (although space most likely is granular) but does aleph-2 then exists in actualized reality ????

Infinity is encountered in daily life as a mental experience of "lacking boundary". Poetic description of a wide landscape, a view of the skies, contemplating about time, they often trigger the word "infinite" (or "endless").

This being remarked, I would also suggest not to rely too much on opinions of mathematicians formulated a century or more ago on the matter of infinity. Before that time, there used to be discussions about existence of infinitesimals, existence of the imaginary unit, and even existence of negative numbers or zero. As Kronecker said in the 19th century, the natural numbers are provided by God, all other number systems are man-made (and a potential subject of dispute).

During the last century, mathematics has become essentially autonomous from reality --reducing such discussions to a technical matter. For instance, natural number arithmetic is completely governed by (Peano's) axioms. The axioms of set theory allow for an object obeying these axioms.They also allow for an elegant extension to integer arithmetic and to arithmetic of fractions. They allow further extensions to real numbers, complex numbers, quaternions, or they allow alternative number systems (e.g., p-adic numbers, computation modulo a prime, etc). Some axioms of set theory (or of other theories) use the key word "there exists ..." (existential quantifier). In mathematical practice, it simply means that you are allowed to name and use an object as described in the axiom. For a simple (and undisputed) example, the second axiom of group theory claims the existence of a neutral element (usually denoted 0 or 1) for the group operation. Similarly, one of the (usual) axioms of set theory claims the existence of a set with a property that necessitates infinity.

Nowadays, "existence" in mathematics does not (primarily) refer to reality. Therefore, discussions on the legitimacy of infinity are either about personal taste or preferences, or about consistence of the corresponding axiom of the theory.

Infinity in the physical universe does not exist. The Planck limit is a clear indication that the physical universe is not infinite.

I have the impression that this thread has already mentioned the issues that are most relevant to the discussion at hand.

1. Many (most?) physicists believe that the universe is essentially finite (limited extend, granularity,...)

2. All computations on physical (and other) problems can be performed on a computer to a satisfactory degree of accuracy.

3. The axiom of infinity in set theory is independent of the other (usual) axioms. It can, in principle, be dropped. Those who wish to keep it do not have to justify this with physical arguments as they are talking only of a "mathematical existence".

4. Theoretical Physics is served well by mathematical models / presentations that are infinite. This is true even at the most elementary level. The theory of Euclidean geometry does not survive undamaged if the plane is replaced by some finite version. The famous numbers "e" (Euler number, basis of natural logarithm) and "pi" loose their meaning. 

In summary, there are good (practical) reasons for dropping infinity from our vocabulary, and there are equally good (theoretical) reasons to keep it. In particular, there  is currently no concrete threat that "infinity" may cause a contradiction (beware, it may well be some day). The Russell paradox of the early 20th century was not caused by infinity, but rather by a naive conception of set theory without regulating axioms.

Dear Marcel,

I don't see that 'e' and 'pi' loose there meaning in a hierarchy of finite mathematical universes modeled by mathematics based on 64 bit numbers, 128 bit numbers, 256 bit numbers, ..... My intuition says me that in such a hierarchical universe we should be able to escape the Gödel conclusion that consitency of mathematics can't be proved by mathematical means. My hope in setting up this question was to learn of arguments that either support this intuition or shatter it. From your contributions I read severe scepticism but not yet a clear statement. So, with this question in mind, it would make sense to continue the discussion and not close it with a nice resummee like yours.

Dear Ulrich,

I am not suggesting to round off this thread. Rather, I wanted to make clear that  pro's and con's about infinity (both in physics and in mathematics) have already been abundantly described. The discussion should also go deeper: how much power is left if infinity is dropped? Is such an approach provably consistent, in other words, does Godel incompleteness fail for finitary mathematics?

Perhaps the issue of "e" and "pi" is a nice one to clarify. My statement that these numbers "loose their meaning" is a bit harsh. You are right to claim that, with growing accuracy, you are approaching these "ideal numbers". But the essence of my statement is, that in a finitary doctrine, it is not fair to claim to approach something that is part of a larger non-finitary system. The old Greek experienced a paradox when a proof showed up that the square root of 2 is not a fraction while apparently it is the length of a certain hypotenuse. In the old world, fractions formed the entire universe of numbers (and for rather good reasons). Similarly, "e" and "pi" seem outside the standard finitary universe.

Some posts in this thread (e.g., Helmut's, Simen's, Leo's) emphasize the value of theory to get insights and that theory seems more demanding for infinity. That is a more general point to worry about.

I wouldn't call my scepticism "severe"; there is too much I do not know. I think your project deserves careful thought.

An intrinsic hierarchy or cardinality of numbers would finitely bound and discretize such infinitesimals as e and pi and sqrt2, etc., just as geometrically in higher dimensions such infinitesimal constants are physically bound or discretized as certain ratios, or hierarchical wholes to parts. 

Cardinality corresponding with Set Theory as a FOUNDATION of mathematics may not be regarded as mathematics itself (Husserlian phenomenological bracketing), and therefore successfully routes outside the purview of Godel's Incompleteness Theorem and Russell's Paradox.

There's maybe another way of putting this Cj :

On the one hand,  infinities exist in non-realized (or non-manifest) reality : such as any infinite series of numbers (of various cardinalities), etc.

On the other hand, the Bekenstein bound, for instance (and other fundamentally similar such limits) puts a cap on infinities within physicality.

So we're in a universe where infinities cannot exist in a manifest rendition, but can exist as a set of unactualized potentialities.

Dear H Chris, Yes, that is precisely and eloquently another way of putting infinities as merely pure abstractions, and thank you also for putting a finger on the Bekenstein bound and similar other limits, which place physical caps on infinities.  However, I contend that even these infinities or any abstractions may be anchored, trumped by, or rooted further (an ultimate and finite, self-similar (ZFC), non-circular, non-tautological, non-paradoxical cardinality) in an even more abstract term (under publication) that is as yet highly concrete and tangible (empirical) and constant in everyday experiences, in every and all descriptions, and in physical reality.

Hi Chris, imagine a universe which is limited. What's beyond the limit? If I approach the boundary of my city I see the "griese area" - sort of wild nature. But what's beyond the boundaries of our universe? Infinite = unlimited = unbounded? So many questions. I better  turn attention to my late breakfast... Only a last remark. In my view the most effective hypothesis about the universe and thus about the world around us is infinity with respect to space & time (and maybe dimensionality), and complexity. WE are finte & limited & bounded, not the world around us. Now the breakfast ...

Helmut, while enjoying your breakfast, you could take a moment to reflect on the possibility that a limited universe does not require a boundary. Just think of surfaces like the sphere and the torus, or non-orientable surfaces like the Klein Bottle and the projective plane for examples in dimension 2. A non-closed universe has no proper  boundary if it takes the form of an open set in Euclidean space or in some surface. Every topologist can tell you about the peculiarity of approaching such a boundary with ever smaller steps. In physical reality, processes in the vicinity, requiring space and time, must get corrupted completely. Your physical presence over there is simply impossible.

Beyond that, the word 'infinite' is a bit content-poor ...

If spacetime is continuous, then the cardinality of the infinity embedded within space time is of aleph-1, which is somewhat low-level. If spacetime is granular, then unless the dimensions of the universe spread out endlessly, there is no actualized infinity in the manifest universe.

In the quantum vacuum though, it is very well possible that higher cardinality "potential" infinities exist, but such are then not actualized

In our human’s process of observation and cognition to universe, we need the concept of “finite” in Mathematics and Physics so we take it, and then the concept of “infinity” unavoidable comes to being. They are there coexisting. No“finite”, no “infinity” and vice versa. So, we have to face, cognize and understand the concept of “infinity” but impossible to say “no” to it or run away from it.

I agree with Geng. Without the concept of infinity, how do we define finite, and vice versa. This is the same argument I have when talking about a fractal Universe. I say the Universe is both fractal and homogeneous. Without homogeneity, with what do we measure fractal and vice versa. 

@Geng

Your mental constitution may force you to think "infinite" and "finite" always together.  Such a psychological observation does not sheed light on the question under consideration. From what was discussed so far, it is obvious that to each infinite model of "the real world" there is a finite one which has the same explanatory power (and vice versa, if the finite version is sufficiently large).  If my intuition is right, there is nevertheless a major difference between the two options:

If a smart pupil asks me: all your physics is finally based on mathematics, and mathematics may be inconsistent. Assuming the contrary is based on mere belief (you read Gödel?) Why then you consider your physical and mathematical stuff more reliable than science fiction? I could anwer: OK, I can't be really sure about the logical consistency of an infinite world model. For any finite subsystem, however, any mathematically derived statement on it is as reliable (also, in fact by the same logic)  as the statement that n+m and m+n  are the same number for any natural numbers n and m.  

Unfortunately, the logic experts in this discussion did not support my intuition.

Dear Lori and Ulrich，

You actually disclosed a good truth: what human science is!

We human exist with our own time and space while Universe and Nature exists with its own time and space; this is one important fact we have to have in our mind all the time. Existing along with human, human science is human’s cognitive product to Universe and Nature------things in our science should be with the property of “human time and space” and “Universe and Nature time and space” so we know how they are from and how they are being in our science.

Thus, we have "infinite" and "finite", "homogeneous" and "fractal", " consistent " and " inconsistent ", "reality" and "model", ….

Exact science strives for optimal reliability, where "optimal" refers to human capabilities. Ulrich's story of the smart pupil is to the point, but the answer to it should be clear. One may wish to achieve the same degree of reliability and certainty in all of our daily concerns. All technical achievements of men -- from prehistoric to modern -- are based on careful observation, experiment, and thought. They all qualify as scientific.

Compare the scientific approach with any alternative "method" of understanding the world. Philosophy gives a general view and offers criticism, but it doesn't build up. Some people deny evolution and dating techniques on religious grounds, combining biased information with dishonest arguments.

Is it really a problem that mathematics has undecidable statements or that it cannot prove its consistence? It troubles me no more than the fact that speed is limited to light speed or the fact that we can't travel into the past. So be it.

Realism dictates that nothing is perfect under the sun.

Dear Marcel,

Does “Infinity” many people talk about here mean “potential Infinity”?

@Geng

I am not keeping track of how many participants in this list refer to which kind of infinity. Most seem to refer to actual infinity. Some people defend actual infinity "with their blood", others keep referring to the supposed finiteness of the universe to drop it. Stll others claim that dropping actual infinity will impoverish the explanatory power of mathematics.

The way I understand Ulrich's question, he would like to get actual infinity out of the way with the least possible mathematical damage. In principle, he is right to answer "no" on his question. There are several axiom systems for set theory that do the job, and computations in physics (and elsewhere) are, de facto, finite-precision. The difficult part of the answer is to measure how much is lost on "explaining power". 

One never knows which inventive new explanations may come up under these limitations. I remember a PhD dissertation  of Timothy Poston, who used finite graph theory to explain a lot of phenomena in traditional calculus and even in relativity. He gave a surprisingly simple explanation for the shortening of rulers at high velocity and the existence of a maximum speed. I have the exact reference in my PhD dissertation (1975) and I must have Poston's dissertation somewhere in my archive (which is rather chaotic since I moved out of my office at my retirement).

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.

The notion of infinitesimals belongs properly to the theory of ordered fields: such a field is Archimedian if each positive element x can grow larger than any given positive element y by adding x to itself sufficiently often. If some positive x fails this test, the field is called non-Archimedian and the failing x is then called an infinitesimal. The name is mere convention (it could as well been called "microscopic" or "quantum"). The word "infinity" does not occur in the entire definition. Yet Cantor's infinity occurs as a consequence (side effect) of "ordered field": such a field (Archimedian or not) must include a full copy of the rationals.

There are many more mathematical concepts which have infinity as a "side effect". E,g,, a nontrivial connected Hausdorff space is infinite. You may work with such concepts without even noticing that it yields infinity. The point is, that these concepts imply infinity without being equivalent to it (an infinite field need not be an ordered one and an infinite space need not be connected or Hausdorff).

Geng, you are calling Cantor's view a "half view" because some other concept implies infinity.

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' because the notions of “infinitesimal”, “infinity” and “infinite” are still confusing as that of Zeno’s time.

Geng,

who loves to be confused will find opportunities even in such innocent topics as Zeno's Achilles--Turtle Race. Who loves clear thinking has to search for other opportunities.

Dear Mr. Ulrich Mutze,

You are right, people have their own chooses by different interests; but love or not, the defects discovered by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race” are there.

It is ok, 2500 years passed in a flash; I just try my best to do something in the rest of my life.

Regards!

Geng

From the physics point of view, Zeno's reasoning makes no sense because of Planck-time (10 E-43 seconds). From a mathematical point of view, Zeno's argument lacks a solid definition of infinite summation (or do you believe that this comes for free?)

Zeno's story of Achilles and the Turtle is a nice heritage from the past, illustrating the dangers of naive mathematical reasoning on reality.

Dear Marcel,

Actually Zeno didn’t create his own definition of infinite, he detected the defects in traditional definition of infinite and made the paradox to disclose the defects and told us that something wrong in the foundation of traditional infinite theory system waiting for us to solve.

His philosophical and mathematical reasoning is really smart; otherwise he was unable to make those paradoxes.

Regards!

Geng

@Marcel

We need to be careful with Planck time as it is part of our present knowledge which is not settled enough to draw strong conclusions. I see not a single reason to drop the notion of infinity, also because we live in an infinitely complex world with infinite scales and beauty. Theologians talk in this context about God. Somehow theology and physics converge here. Ulrich, you talking about a fraction of math where possibly this notion is dispensable. In general it is not. The digital world is only a tiny fraction of the WHOLE THING. HZB

I have been studying fractals for dozens of years now and I can tell you for certain, although fractals have infinite potential, it is the limit to the digits of precision of the computer that prevents a true infinity. The universe has a similar limit, that is the Planck limit.

@Helmut,

Ulrich, Helmut,

Abstraction could also be described as neglection of irrelevant details (the judgement of irrelevance being left to the wisdom of the neglector). When used with wisdom, it is a positive kind of ignorance.

I fully agree that there are many useful abstractions unrelated with infinity. The branch of mathematics that is most involved with finiteness is Discrete Mathematics, including Graph Theory and Combinatorics. It is beautiful, it is relevant, and it is clever. Also, Finite Fields (in its simplest form, computing modulo a prime) have gained much interest with the advent of computers. It is also used in coding,  cryptography and statistical simulations with random number generators.

Finite mathematics is certainly not one of mathematics lesser children!

@ Helmut and Lori,

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)”.

It is just with limit idea and limit theory that make us possible turn infinite things into finite things and cognize the infinite world from quantity aspect.

@Marcel & Geng: Thanks, very helpful comments!

@Lori: Dear, the Planck limit is still a speculative item. It is dangerous to derive fundamental answers from such a still open or fuzzy concept.

@Ulrich: We possibly should make a difference between (1) algebraic+logical calculations and all what a computer can do (discrete math as mentioned by Marcel), and (2) Math, which includes calculations but goes well beyond.

Abstraction is a qualitative feature which defines a difference between human beings and machines. You mention Euler and Gauss: Euler introduced the frictionless fluid (viscosity -> 1/infty), which is actually sort of "inert Euclidian geometry", i.e. pure abstraction. Take calculus: we often are forced to apply boundary conditions at x=infty to get an analytical solution etc. Only in a programmer's world infty seems to be obsolete, but there's also life outside the programmer's world...

You may ask 'Why do we need analytical solution of diff eqns?" if we have computers? Answer: If you want to sell a numerical solution method you need a sales argument for the customer, e.g. the accuracy of the solution relative to the exact solution, which is an analytical one. Accuracy (a vector quantity) acts like horespowers or fuel consumption in cars.

Accuracy is based on analytical solutions of a differential equation. E.g. the Gaussian solution of a parabolic time-dependent heat equation assumes boundary conditions at infinity etc. etc. - alway infty comes in, even in the very practical work of selling computer codes or programs.

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

@Geng: Unfortunately today we know that Zeno's paradoxes are simple fallacies and mainly due to the fact that Zeno did not yet know the concept or notion of velocity as a relative invariant of motion. Thanks to Weierstraß and all the others who contributed today we know better. (I have had the honor to work a number of years in the Weierstraß-Institut für Mathematik der Akademie der Wissenschaften der DDR, in Berlin, Hausvogteiplatz.)

@ Helmut

In fact, the defects disclosed by Zeno’s infinitesimal related paradox family of “Achilles--Turtle Race” are the logical contradictions between so called “potential infinity” and “actual infinity” theoretically and practically; it has nothing to do with the concept or notion of velocity.

According  to my studies, the typical modern version of this Zeno’s Paradox is the Paradox of Harmonious Series.

@Geng: You're generally right, but in the specific case of Achilles vs. turtle velocities matter. Anyway, thinking about infinities is the core of mathematics. Have a nice evening, HZB

Dear Dr. Ulrich Mutze,

Yes we need the concept of Infinity. If we look the natural behaviors of some elements we realize the importance of Infinity particularly, when the behavior follows exponential decay curve.