Natural Philosophy

Scientific Mathematics / Mathematics as a Science

"Reality is that which, when you stop believing in it, doesn't go away" (Philip K. Dick)

  1. Introduction
  2. Consistency and Madness
  3. Propositional Logic
  4. Mathematical Identity
  5. Identity <> One and only
  6. ZFC Killer Axiom
  7. Members = Classes
  8. Elementary Partitions
  9. Infinitum Actu Non Datur
  10. The Physics of Infinity
  11. Balls in a Vase
  12. Political Economy of Sets
  13. Implementable Set Theory
  14. Function = Mapping + Time
  15. Brouwer's Continuity Theorem


When talking about the
Foundations of Mathematics (: excellent overview !), two main directions of thought shall be distinguished: It's not my purpose to repeat here the pro's and con's of the Formalist versus the Intuitionist doctrine (though personally I have a slight bias towards my fellow countryman L.E.J. Brouwer). From the point of view of a Physicist - yes, I am one by education - what really counts is the applicability of the mathematics they provide us with, irrespective of any underlying philosophy. But it should be emphasized from the very start that applicability is the power to be applied, which is not the same as actually being Applied !
I think that nobody can deny that Physics is the most advanced science in our society nowadays. (It's the most dangerous and the most irresponsible science as well, but that's another problem). Physics is also the science that makes extensive use of all kind of mathematics. Therefore I find it strange that no mathematician has ever asked the opinion of a physicist in these matters. Is it because of Einstein's sneering comment on the controversy as the battle between the frog (Hilbert) and the mouse (Brouwer) ? Let it be stated here very firmly that the outcome of a battle in Mathematics might be important indeed, also for physicists like Albert Enstein. But on the other hand, what makes Mathematics so "independent of reality" that mathematicians shouldn't be bothered, in any respect, about developments in other sciences, like (for example) Physics ? Now I am asking you, mathematicians, is it truly reasonable (you are reasonable, aren't you ?) to assume that Elementary insights of Physicists can have no impact, at all, on the Foundations of Mathematics ?

Consistency and Madness

Go to a mental hospital and I'll bet you will meet people there who call themselves Napoleon, or Jesus Christ, or Elvis Presley, or Albert Einstein.
For the sake of simplicity, let us take the man with the Einstein complex. Now suppose that you take such a man apart and you decide to talk with him, in order to convince him that Albert Einstein is dead and buried. And that his real name is Johnson. And that he is just the man around the corner. No genious at all. Do you think you are going to be successful?
You talk to him for more than one hour, trying to convice him that he should give up his picture of the world. At last, you ask him if he has understood your arguments. I'll bet his answer will be like the following: "Yes, of course I understand ! Because Albert Einstein is a genious, so he can understand anything. And I am Albert Einstein."
Now replace "Einstein" by "mainstream mathematician" and you're done.
Nobody can deny that people in a mental hospital have a consistent picture of the world. Now the good news of consistency is that, once you are on the right track, you will remain on the right track. The axiom system for Euclidian Geometry is a good example of this manner of being consistent.
But here comes the bad news. Once you are on the wrong track, you will always be on the wrong track.
There is no way to tell a mathematician that the axioms of the Zermelo Fraenkel / axiom of Choice (ZFC) system are "not good". Because then he will defend himself as our would-be Einstein character did. All of his arguments will be consistent with the system he believes in. He wants to remain in his vicious circle, safe and well. Calling everybody else a "dude" and a "crank" and a "zealot". And there is no way out. There is no cure for his madness. Because mathematics is what mathematicians do.

Propositional Logic

Quoted without permission from the GetReal presentation: Attempts to reduce modern mathematics to logical tautologies have failed miserably in practice and may have been doomed from the start in principle.
It may be said that Propositional Calculus hasn't been successful, at all, in replacing common language. Not with reasoning in the exact sciences, not even with reasoning in mathematics. Mathematical Logic has been doomed from the start, indeed. In order to understand the sequel, it's necessary to comprehend the difference between the so called material implication and formal implication in the first place. Therefore, in case you still don't, click this hyperlink, which is also in the header of the section.
Now, quite contrary to Alfred Tarski's holy belief, the obvious divergency between the usage of the phrase "if ... , then ..." in ordinary language and its usage in mathematical logic may be well rooted at the cause of the abovementioned failure of Mathematical Logic. I have always wondered how so many of these logicians have managed to become such excellent linguists too, with such a steep learning curve. Because otherwise I cannot understand how they have the nerve to tell us that "There is no phrase in ordinary language which has a precisely determined meaning". Especially, how they have the guts to claim that "if ... then ..." has not a precisely determined meaning in common speech. It should be emphasized here that "common speech" refers to all languages in the world, not only to English ! And indeed, all discrepancies between the "if ... then ..." in common language and the "if ... then ..." in mathematical logic seem to be of a very much alike nature everywhere. Tarski's book has been translated in several languages, Dutch for example. There hasn't been any apparent need to make significant changes in the text, though, due to any logical peculiarities of the language in question. So all of the discrepancies between material and formal implication (how come such misnomers !) holds for all of the languages the book has been translated in. Thus it seems that mathematical logic is incompatible, not only with English, but with every other human language in the world, even with ancient languages, like Latin. (I know this, because I have studied Latin and Greek). However, people have employed common logic for centuries. Whole civilizations have been built up with the help of common "if ... then ..." constructs. Could it be, perhaps, that logicians, like Alfred Tarski, have studied no linguistics and no foreign languages, at all ? (No, it seems that Alfred has enjoyed a decent education). Or do they exhibit this disdain for the language of common people because they have actually failed to make a proper abstraction of the "if ... then ..." phrase? And, with their arrogant utterances, they are only trying to defend their childish "new language" against the intuitively well established logic of common speech ?
Despite of all this, Tarski has been close to a solution of the problem: The concept of formal implication perhaps has not been made completely clear, but at any rate, it is narrower than that of material implication; every meaningful and true formal implication is at the same time a meaningful and true material implication, but not vice versa. Look what he says: the implication of common language is more restrictive than the implication of mathematical logic. Now, instead of looking upon common speech with contempt, better ask why the implication of common language is more restrictive. Because, as many things in nature, it could have a good reason. I could easily think of something like efficiency, for example. To put it in other terms: the implication of mathematical logic may be too general, when compared with the implication of common language. Which IHMO is the only correct conclusion possible.
My conclusion that the mathematical implication may be too general is supported by several other observations. Consider, for example, the IF ... THEN ... ELSE ... construct in our traditional programming languages. And simply take notice of the fact that there hasn't ever been any controversy about it. Yet - as is the case with mathematical logic - we are talking about artificial languages ! The main difference is that, apparently, a proper abstraction of the implication has been implemented in those languages. (And the question arises whether such a thing could be mimicked within Propositional Logic).
At the time I finished my Physics curriculum, at the University of Technology in Eindhoven (the Netherlands), two famous professors were supposed to be involved with my mathematics education: N.G. de Bruijn (who is not a relative of mine) and E.W. Dijkstra. I have met both of them personally. Both personalities have produced significant contributions to mathematics and Computer Science. E.W. Dijkstra may be considered as one of the founders of the (in)famous "correctness proofs" for computer programs. He finds that the correctness of a computer program should be established by a mathematical proof. On the other hand, N.G. de Bruijn et al. developed a computer program, called AUTOMATH, for checking the correctness of mathematical proofs. He finds that a proof in mathematics should be formalized in such a way that a machine can check the correctness of it. It is evident that there is kind of a vicious circle here: let AUTOMATH check the correctness of a program correctness proof according to Dijkstra and let a correctness proof according to Dijkstra check the correct performance of AUTOMATH. Apart from that, it is clear that a mathematical proof that has been found to be correct with AUTOMATH does not have to be correct according to the logic of common language. That is because we found that propositional logic is too general, hence a proof which is correct in propositional logic doesn't have to be correct within the logic of common speech. (Here it is assumed that the AUTOMATH program employes propositional logic. I don't know if such is actually the case).
The question of whether a computer can think is no more interesting than
the question of whether a submarine can swim. 
- E. W. Dijkstra 

Mathematical Identity

Several silly stories about Perpetuum Mobilae appeared in that newsgroup ('sci.skeptic'), kicking in an open door. Virtually nobody seems to be skeptic about established science. Well, certainly I am more skeptic about the latter than about the former. Simply because "true science" is far less open to dispute than quackery.

I would like to start with the science of Mathematics. And my first question is: which symbol is the most frequently used in mathematical formulas ? I am pretty sure that it is the sign for "equality" or "identity"   =   . Now I think that it is impossible to develop Predicate Calculus, as a part of mathematical logic, without the whole notion of identity. But let us assume for a moment that it is possible nevertheless. Then we can find a kind of mathematical definition in 'Principia Mathematica' by Russell and Whitehead, chapter 'Identity ':

   ( a = b )  :<->  ∀P:  P(a) <-> P(b) .

a equals b means that, for all properties P: P is a property of a if and only if
P is a property of b .

This sounds reasonable. Consider, however, the expression ' 1 = 1 ' . Then, the ' 1 ' on the left of the equality sign is not equal to the ' 1 ' on the right of the equality sign, because the property 'being on the left' is obviously not equivalent with the property 'being on the right'. Therefore we conclude to a paradox: 1 <> 1 .
Therefore Identity is not properly defined by the logic in 'Principia Mathematica'. But now we are in trouble; I have never seen another valid definition of Equality, the equals sign ' = ' , the most important mathematical concept in existence !

This is not the end of the story, though. So called "Equivalence Relations" are supposed to share the following properties:

      a ≡ a   ;   a ≡ b  ->  b ≡ a
    ( a ≡ b  and  b ≡ c )  ->  ( a ≡ c)
Quoting a (Dutch) textbook, equivalence relations should be conceived as a "generalization" of "ordinary" equality. But tell me: how can something be generalized, if it is not even properly defined itself ?

Time has come to give my own opinion.
There is no such thing as an "absolute" identity. Every identity is only in some or in many respects. Equivalence relations cannot be distinguised, at all, from "ordinary" equalities. Just replace ' ≡ ' by ' = ' . What's in a word anyway ?
Distinguishing "equivalence relations" and "definition by abstraction" from "common identity" and equality are merely good examples of how to make mathematics unnecessarely complicated.


Jan Willem Nienhuys:
This comes from using a formal language (in this case the language of Principia) to talk about itself, namely about the syntactical structure of the formulas. It is known that doing so always risks contradictions. Actually, I doubt whether you can formulate 'being on the left' in the language of the Principia, anymore than you can refer to the color or the brand name of the ink used to print the formula, or the font of the letters used.
Similar games can be played with the equality a = b, to prove that b is the first letter of the alphabet or from Venus = Aphrodite that Venus has 9 letters.

John Franks:
Consider, the expression

          Hans de Bruijn = Hans de Bruijn
Then, the Hans de Bruijn on the left of the equality sign is not equal to the one on the right of the equality sign, because the property 'being on the left' is obviously not equivalent with the property 'being on the right'. Therefore we conclude the original poster must have been (at least) two people.

Doug Merritt:
The spot where you went wrong is in considering 'being on the left / right' to be valid predicates, for in terms of algebra, they are not. In fact, this is a specific instance of a very general confusion about mathematical systems, which is that the language used to describe a mathematical system generally does not follow the same rules as the system being described.
Formal axiomatic systems (the ultimate in mathematical rigor) use a meta language to describe the language of the system being set up.
This dual language system has been explored very extensively, and there are some very interesting results, such as the impossibility (I believe it's been proved) of using a single meta language to define both itself and the usual axiomatic systems of analysis.

Richard O'Keefe:
Do we have to drag out the type / token / quotation stuff all over again? In the equation

          1 = 1
          ^ ^ ^
          a b c
there are three parts, labelled a, b, and c. a labels a token belonging to the type '1', b labels a token belonging to the type '=', and c labels a token belonging to the type '1'. a and b clearly label different tokens (as de Bruijn observes), but the equation is not mentioning those tokens but using them to refer to the number 1. If de Bruijn's argument were valid, we would never be able to talk about anything at all. (There used to be another occurrence of that name in this sentence but my deleting it didn't delete de Bruijn himself.)

Okay, that's fair enough.
Of course, I have been well aware of this "confusion" between the tokens and the meaning of the tokens. But let's carefully rephrase from the 'Principia ':

   ( a = b )  :<->  ∀P:  P(a) <-> P(b) .
Am I blind or what ? Did nobody read the ' ∀ ' in that statement ? I did nothing else that take that forAll quite litterally, just as it is meant - without doubt - by the authors themselves. And now everybody comes to me and says that it is All, but except being on the left, except the brand name of the ink, except the font of the letters used, except the rank in the alphabet. With other words: except almost everything ! Therefore it is not a forAll, at all !
My point is this. I wouldn't have made any trouble, if Russell and Whitehead (and many mathematicians) would have been humble enough to recognize that there simply does not exist such a thing as "all properties ". A definition like the following would have been more than sufficient, instead:
   ( a = b )  :<->  Some P:  P(a) <-> P(b) .
Now give me a enumeration of any properties that you want to be allowed for the purpose of identifying the "meaning" of the symbols used (: Pattern Recognition techniques ?) And I'll be no longer your opponent.

Useful remark: the set of Properties needed for Identification basically will be finite. Further thinking along these lines reveals that:

Identity <> One and only

With other words: being Identical is not the same as being One and only one.
Seven is seven. But: seven apples is not the same as seven pears. Zero is zero. But: having no bread is not the same as having no water.
It goes much deeper than this, though.
In quantum mechanics, there is the concept of
Identical Particles. Quoted without permission from another website:
One of the fundamental postulates of quantum mechanics is the essential indistinguishability of particles of the same species. What this means, in practice, is that we cannot label particles of the same species: i.e., a proton is just a proton; we cannot meaningfully talk of proton number 1 and proton number 2, etc. Note that no such constraint arises in classical mechanics.
But the latter is not true. Suppose that we have two identical pictures on a web page, as implemented by:
    <TD WIDTH=410><IMG SRC="zoeken01.jpg" WIDTH=400></TD>
    <TD WIDTH=410><IMG SRC="zoeken01.jpg" WIDTH=400></TD>
Then the situation is exactly analogous as with the identical particles in QM. If the pictures are labeled, then we cannot meaningfully talk of picture number 1 and picture number 2. Suppose, namely, that the picture on the left is labeled as 1 and the picture on the right is labeled as 2. Exchange the two pictures. Since the pictures are identical, it's impossible to tell whether they are interchanged or not. Yet there are two pictures instead of one. They are identical, but they are not one and only one thing.

The above has immense consequences for mathematical reasoning. As an example, let's recall the following theorem: there exists one and only one empty set. Which is not true. Actually, there do exist many empty sets. And they are all identical. But they are not one and only one set. Another example, with more profound consequences, is the set of all naturals. All sets of natural numbers are identical. But this does not mean that there is only one such a set. Another way of looking at this is that the mathematical abstraction of the set, or maybe rather the class of all naturals, has many instances, to speak in terms of Object Oriented Programming. Let us reconsider now the theorem that the set of all natural numbers has the same cardinality as a proper subset of itself, namely the set of all even natural numbers. This is commonly visualized as follows:

         1  2  3  4  5  6  7  8  9 10 11 12 ...   
         2  4  6  8 10 12 14 16 18 20 22 24 ...   
Sie wissen das nicht, aber sie tun es (: Karl Marx). They don't know it, but they do it. It is seen here that the set has been splitted already in two: (1) the set of naturals and (2) the set of even naturals. This has as a consequence that the set of even naturals is no longer considered as a subset of the set of all naturals. Thus maybe the set of all even naturals is a proper subset of the set of all naturals, but it is no longer considered as such. Herewith, a highly counter-intuitive aspect of Cantor's theory of Cardinals is removed, a great deal. Because now it says that there exists a set of all naturals. And another set of even naturals. And there is an equivalence relation between these two sets; therefore they have the same cardinality. The set of even naturals is just the set of all naturals with a "re-named" unit element for counting, called 2 instead of 1.

Παντα ρει, ουδεν μενει (Panta rhei, ouden menei), to be translated as: "Everything flows, nothing remains" (Heraclitus of Ephesus, 540-480 BC). Thus one might even argue that there isn't a thing which is one and only one. Let time be attached as a label to anything. Then it is obvious that anything is changing with time. Meaning that anything can always be considered as arbitrary many identical instances. When it comes to (re)conciliation of Mathematics with Physical Reality, I think this is one of the main issues to be considered. The key concept being that the mathematical abstraction is like an OOP class - quite close: it's a set, anyway. And physical reality is like a bunch of instantiations of that class.

ZFC Killer Axiom

As a matter of Natural Philosophy, we have established that, if x is a member of X , then certainly there is not a physical reason why x should not be, at the same time, a part (i.e. subset) of X . Since I am a physicist, not a mathematician, I cannot even imagine an element which is not a subset as well. In addition, being a part of something always seems to be more general than being an element-ary part of something. Can someone please stand up and explain me how to acquire a measuring device that can ever distinguish a single x from the set { x } containing that single element x ? I am rather certain that no such a device can be found in the entire Universe ! Therefore the following
additional axiom (11) for Zermelo-Fraenkel Set Theory (ZFC) will be proposed:
            ∀ x : x = { x } 
In words: the box {} is not a part of the set. An apple cannot be distinguished from the set which is containing only that apple as a member.
But anyhow, with the 11th axiom, it immediately follows that:
     ∀x,X : (x ∈ X)  ⇒  ({x} ⊆ X)  ⇒ (x ⊆ X) 
With other words - take my advice - people should better: If x is a member of X then x is also a subset of X . Quoted from a (lost) reference called "The Ordinals": Essentially, every element of a Transitive set is a subset of that set . Thus: in our theory, every set is transitive .

Look what I am doing. Being a civilized debater, I whole-heartedly agree with those great achievements of Set Theory ;-) I have embraced all 10 axioms of ZFC. But as a physicist, unfortunately, I have to build theories which are in close agreement with Physical Evidence. Therefore I was forced to augment the ZFC system with that tiny additional axiom. The resulting system could be called "ZFC augmented" or: ZFC+ . I am very well aware of the fact that ZFC+, due to its 11 axioms instead of 10, will yield less valid theorems than ZFC. (Geez, did I do that on purpose ?) Provided that our system is consistent - because if such is not the case, there will be no theorems at all ! If you cannot accept these terms, you'd better quit this forum, here and now. For the rest of us, the time has come to lean back and see what happens if the 11 axioms of ZFC+ interact with each other.

Oh well, this has become a much more dramatic excercise than I first thought. A (now obsoleted) web-page has given rise to a heated debate about  x = { x }  in sci.math.
Jesse F. Hughes and Chas Brown proved the following, with common set theory: there is only one set in my ZFC+ universe.

Lemma.   ∀ x , y : ( y ∈ x ) ⇔ ( y = x )
Proof (as quoted from the thread).

Roughly speaking, (Ax) x = {x} implies that if y in x, then also y in
{x}; but {x} is a set with only one element, so it follows that we must
have y = x.

The above uses the standard axiom that says that two sets are equal if
and only if they have exactly the same members; i.e. the axiom that
states that (Ax)(Ay) ((x=y) <-> (Az) (z in x <-> z in y)).

Cheers - Chas
The latter remark by Chas may be essential.
Anyway, with help of the Lemma, the proof of Jesse's Theorem is easy:
Weak pairing) (A x)(A y)(E z)(x in z & y in z) 

Now, let's see what we can prove with that plus Han's Brilliant 
Discovery that x = {x}, rewritten with definitions expanded. 

(HBD) (A x)(A y)( y in x <=> y = x ) 

Put those together and what happens? 

Let x and y be given and let z be a set containing both x and y (from 
weak pairing).  Applying HBD, we see: 

  x in z <=> x = z       and         y in z <=> y = z. 

But since x *is* in z and y *is* in z, we see that x = z = y, and thus 
x and y are equal. 

Hence, weak pairing + HBD proves that there is only one set in the 
Thus ZFC+ contains only one set. But it gets even worse. As quoted from the same article:
It is trivial to prove that augmenting ZF with (A x)x = {x} is 
inconsistent.  First, prove that the empty set exists -- a trivial 
theorem in ZF.  Then we have by your axiom that {} = {{}} and hence {} 
is an element of {}. But by definition of {}, we also have {} is not 
an element of {}. 
Thus there exists only one set, which then must be the empty set, because we can prove in ZFC that the latter exists.
But then we can prove with  x = { x }  that the empty set is both empty and non-empty.
Which completes the proof that ZFC+ is inconsistent and has shriveled into nothingness.

Thus  x = { x }  indeed blows up the whole of ZFC ! That's why we shall call   x = { x}  the ZFC Killer Axiom.
Physicists would say that ZFC is a highly unstable system, since it already blows up when a seemingly innocent change in its foundations is being made.

Blows up !? Han, what are you doing !? What I am doing ? I'm doing just Physics !
The fact that the eleventh axiom is according to physics and that ZFC plus that eleventh axiom shrivels into nothingness doesn't hurt me so much, since the alternative would be something much worse. Leaving these mathematical systems intact simply means allowing that anomalies w.r.t. the real world certainly are to be expected. Provided that  x = { x }  is a sensible axiom indeed, physically speaking, then we have shown that ZFC is contradictory to real world experience. But this would mean that the Axioms of ZFC may be contradictory to the Laws of Nature. Hence the foundations of mathematics may be incompatible with the foundations of physics. Therefore: No Mathematics (with ZFC) XOR No Physics. If this is really our choice, then I would say: make up your mind !

Even more shocking is that the blowup occurs at such an elementary level. As Jesse F. Hughes has analysed correctly:

even though it is inconsistent with the minimal subtheory of ZF 
defined by: 

(1) Pairing axiom: for any x,y, there is a set containing just x and y. 

(2) Non-triviality condition: There are (at least) two distinct sets. 
Note that { x } is a set such that (a) x is in { x } and (b) if y is 
in { x } then y = x.  This is what { x } *means*. 

Let x and y be distinct (by (2)).  Form the set {x,y} (by (1)).  Then 
by your axiom {x,y} = {{x,y}}.  By (1), x is in {x,y} and hence x is 
in {{x,y}} (Note: this is not an application of extensionality, but 
just the logical axiom of substitution.).  Similarly, y is in 
{{x,y}}.  But by (b), we see that x = {x,y} and y = {x,y}, so x = y. 
With other words, if we replace the abstract universe by a universe of bullets:
  1. For any bullet x , the set {x} containing this bullet is just the bullet.
  2. For any two bullets x,y , there is a set containing both: {x,y}
  3. There are at least two distinct sets of bullets: x and y .
Combining (1) and (3) gives that there exist at least two bullets. And with (2) there exists a set {x,y} containing both.
Now where is the paradox that killed ZFC ? Where is it gone ? ( Sometimes I don't understand a thing about set theory )-:

Last but not least, I apologize for eventually having (ab)used the mental capacity of my fellow debaters in 'sci.math', for the purpose of breeding my own offspring.

As quoted from The Universe of Discourse: Frege and Peano were the first to recognize that one must distinguish between x and {x}.

Members = Classes

Let's consider Set Theory as it still is nowadays. What is the definition of a set ? According to Georg Cantor, the founder of set theory himself, a set is:
Eine Zusammenfassung von bestimmten wohlunterschiedenen Objecten unserer Anschauung oder unseres Denkens (welche die Elemente genannt werden) zu einem Ganzen. Thus Set Theory is based upon the "primitive notion" that something can be an element / a member of a set. A set is defined by its elements / members.

The proposition   " a is a member (or element) of A "   is denoted as: a ∈ A .
Let A and B be sets. The proposition   " A is a subset of B "   is denoted as: A ⊆ B .
We are ready now to "understand" the following transla(ted quota)tion from a (Dutch) mathematics text book:
Essential for the theory of sets is the distinction between set and element. Never mix up ⊆ and ∈ , a and { a } .
(S.T.M Ackermans / J.H. van Lint, 'Algebra en Analyse', Academic Science, Den Haag, 1976)

But, for example, if F := { S , T } , S := { a , b , c , d } , T := { a , c , e } , why then are S and T elements of the "set" F ?
Since S and T have the elements a and c in common: so they are not "wohlunterschiedenen Objecten", if we strictly adhere to Cantor's description.

And, on the contrary, if x is a member of X , then certainly there is not a physical reason why x should not be, at the same time, a part (i.e. subset) of X . Since I am a physicist, not a mathematician, I cannot even imagine an element which is not a subset too. In addition, being a part of something always seems to be more general than being an element-ary part of something. Being a subset seems to be more general than being a member. That is, we are tempted to conjecture the following theorem:

     ∀x,X : (x ∈ X)  ⇒  (x ⊆ X)
But leave this for a while as it is. And let's continue rephrasing standard knowledge. The empty set is denoted as Ø .
If A is a set, then the set of so called classes { x,y, ... } is called a partition of A , iff the following conditions hold:
  1. x ≠ Ø , a class is allways an non-empty subset;
  2. (x ≠ y) ⇒ (x ∩ y = Ø) , two different classes are disjunct;
  3. x ∪ y ∪ … = A , all classes together make up the partition as a whole;
    a partition is completely determined by its classes.
Considering partitions, what do you think of the following "analogy": It is thus clear from this presentation that there is no "essential" difference between the classes in a partition and the elements in a set. In mathematical terms: elements and classes are isomorphic. In plain English: classes and elements denote the same kind of thing.

              MEMBERS = CLASSES

Now it's a well known fact that partitions are closely related to equivalence relations. And the members of a set are closely related to "ordinary equality". Meaning that the above proposed system is quite consistent in this respect: just read the paragraph about Mathematical Identity again.

An example from group theory: Different rotations, say about 45, 90, 180, 360 degrees can be taken together to the general class of rotations. But these rotations are themselves already "defined by abstraction": rotations carried out in Eindhoven or New York, taking an hour or a second, done by me or by Jan Willem Nienhuys.
    90o rotation = 360o rotation (: equivalence class)
    is not essentially different from:
    90o in Eindhoven = 90o in New York (: identity)

Elementary Partitions

We can think now of a Top-Down approach to Set Theory, instead of the usual bottom-up approach. In many cases, Cantor's "primitive notion" of the members in a set is not a relevant concept anyway. It has been established in practice that a kind of "set theory without the elements" is perfectly possible. Support for such a set theory - void of "elementalism" - is found, for example, in '
Constructive Solid Geometry ' (CSG), where simple geometrical base shapes are put together with Boolean combinations such as intersection and union. Nobody is really interested in the "members" (read: single points) of the models involved here !

The top-down approach has to be initialized with the axioms of Boolean Algebra, instead of ZFC. So let's start with Boolean Algebra. In order to enable the bootstrap, some rudimentary (naive) set theory may be assumed as well (as in the MathWorld page), but it is is not strictly necessary. Anyway, here goes.

A Boolean Algebra is defined for objects   x , y , z , ...   together with two binary operations  ∩  and  ∪  and a unitary operation  '  such that it satisfies the folowing axioms:

  1. Idempotent laws:   z ∩ z = z   and   z ∪ z = z
  2. Commutative laws:   y ∩ z = z ∩ y   and   y ∪ z = z ∪ y
  3. Associative laws   x ∩ (y ∩ z) = (x ∩ y) ∩ z   and x ∪ (y ∪ z) = (x ∪ y) ∪ z
  4. Distributive laws:   x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z)   and x ∪ (y ∩ z) = (x ∪ y) ∩ (x ∪ z)
  5. Universal bounds Ø and I :   Ø ∩ z = Ø   and   I ∩ z = z   ;   Ø ∪ z = z   and   I ∪ z = I
  6. Complementation:   z ∩ z' = Ø   and   z ∪ z' = I
  7. De Morgan's laws:   (y ∩ z)' = y' ∪ z'   and   (y ∪ z)' = y' ∩ z'
  8. Double complementation:   (z')' = z
I'm aware of the fact that the above axiom system is somewhat redundant. But the most important thing is that it is nevertheless consistent. And that everything necessary is at our immediate disposal.

Since in our Top-Down approach there are no elements yet, we cannot speak about a Set, let it be about its Members. The term "Boolean Object" is suggested instead, for a "set without members". In order to furnish Boolean Objects with the properties of a "real" set, we first have to establish what "being a part of" means.
The following:   (A ⊆ B) ≡ (A ∩ B = A) . Or equivalently:   (A ⊆ B) ≡ (A ∪ B = B) .
Here the right hand sides are defined by the axioms of Boolean Algebra for the objects A and B.

Hereafter, being a Partition is defined in the same way the mainstream theory of classes and partitions is set up.

Definition. A partition of a boolean object V is a (naive) set U of boolean objects W with the following properties:

  1. ∀ W ∈ U : W ≠ Ø . The members are non-empty.
  2. ( ∀ W1 ∈ U ) ( ∀ W2 ∈ U ) ( (W1 = W2) ∨ (W1 ∩ W2 = Ø) ) . Two different members are disjoint.
  3. ∪ W ∈ U = V . All members together make up the object as a whole.
But, ah, a picture says more than a thousand words. The object V is in blue. The inequality is for the partitions.

No matter how you subdivide that piece of cake, it remains the same piece of cake. But the partitions are different.
Or:   9 = 5 + 4 = 3 + 3 + 3 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 . It's still the same number, but differently composed every time.

Let's proceed. Once upon a time, Euclid said the wise words: a point is what has no part. And who am I to deny it.
Definition. A boolean object is called a point iff there doesn't exist another boolean object being a proper part of it.
That is, if E is a point then there doesn't exist any P with  P ⊂ E   and vice versa.
Definition. An Elementary Partition (EP) is a partition where all members are points.

Example. Another basic axiom of Boolean Objects could be that there exist at least some of them. Let these objects be named a , b . And let there be no other objects in our universe of discourse. Theorem : then   { a' ∩ b , a ∩ b' , a ∩ b }   is an elementary partition of   a ∪ b  provided that   a' ∩ b ≠ Ø   ;   a ∩ b' ≠ Ø   ;   a ∩ b ≠ Ø   .


  1. (a' ∩ b) ∩ (a ∩ b') = (a' ∩ a) ∩ (b ∩ b') = Ø ∩ Ø = Ø
    (a' ∩ b) ∩ (a ∩ b) = (a' ∩ a) ∩ (b ∩ b) = Ø ∩ b = Ø
    (a ∩ b') ∩ (a ∩ b) = (a ∩ a) ∩ (b' ∩ b) = a ∩ Ø = Ø
    Thus all members are disjoint.
  2. (a' ∩ b') ∪ (a' ∩ b) ∪ (a ∩ b') ∪ (a ∩ b) = (a' ∩ (b' ∪ b)) ∪ (a ∩ (b' ∪ b)) = (a' ∪ a) = 1
    Now form:   (a ∪ b) ∪ ((a' ∩ b') ∪ (a' ∩ b) ∪ (a ∩ b') ∪ (a ∩ b))
    Due to the above, this is equal to:   (a ∪ b) ∪ 1 = (a ∪ b)
    On the other hand, due to de Morgan's law: (a' ∩ b') = (a ∪ b)'
    Therefore:   (a ∪ b) ∪ ((a' ∩ b') ∪ (a' ∩ b) ∪ (a ∩ b') ∪ (a ∩ b)) = ((a ∪ b) ∪ (a ∪ b)') ∪ ((a' ∩ b) ∪ (a ∩ b') ∪ (a ∩ b)) = (a' ∩ b) ∪ (a ∩ b') ∪ (a ∩ b)
    Thus the members make up the whole object  a ∪ b .
  3. We cannot form any proper parts of (a' ∩ b) , (a ∩ b') , (a ∩ b) , if only the objects a and b are given.
Things become even more complicated if we consider the possibility that some of the point-like members may be empty:   a' ∩ b = Ø   ;   a ∩ b' = Ø   ;   a ∩ b = Ø   .
There are eight (8) possibilities in total:

  a' ∩ b = Ø     a ∩ b' = Ø     a ∩ b = Ø   Elementary partition of   a ∪ b  
False False False { a' ∩ b , a ∩ b' , a ∩ b }
False False True { a , b }
False True False { a , a' ∩ b }
False True True { b }
True False False { b , a ∩ b' }
True False True { a }
True True False { a } = { b }
True True True Ø

The above procedure of forming minterm elements as the points in an elementary partition can be carried out with any number of Boolean Objects.
The theory of minterms is employed extensively with the design of logical devices. A lucid representation of them is the so-called Karnaugh Map.
An excellent exposure (and much more) can be found in: H. Graham Flegg, 'Schakel-algebra', Prisma-Technica, translated from: Boolean Algebra and its Application, Blackle & Son Ltd. London and Glasgow 1965.

Example. Let U and V be two different partitions of the same set A , where U = {a,b} and V = {c,d} . And the only boolean objects given are in {a,b,c,d}.
Question: What is the elementary partition E of A ? Answer: E = { a ∩ c , b ∩ c , a ∩ d , b ∩ d } , provided that these members of E are non-empty.

Example. The elementary partition E of an arbitrary set A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , ... } is given by E = {{1},{2},{3},{4},{5},{6},{7}, ... } .
If x is a member of A : x ∈ A , then {x} is a subset of A : {x} ⊆ A , but {x} is a member of E : {x} ∈ E .

Example. The set X itself is a member of the trivial partition T of X , consisting of the set as a whole : T = {X} .
An EP set X can be an element of itself, but only if the trivial partition {X} of X is considered. Then X ∈ X and X = {X} .
If the trivial partition of X is excluded, then X can not be an element of itself (: cleans up Russell's paradox).
(Note. The resemblance between a set and the trivial partition, associated with it, explains much of my "confusion" :-)

Example. An "ordered pair" (a,b) may be defined in common Set Theory by {{a},{a,b}} .
Such an ordered pair is not a partition of any set, because two different elements in a partition should have nothing in common. But: {a} ∩ {a,b} = a .
Theorem. An "ordered pair" (a,b) can not be defined by   {{a},{a,b}} , because two different elements in an EP set should have nothing in common.
Proof:   {a} ∩ {a,b} = a ∩ (a ∪ b) = (a ∩ a) ∪ (a ∩ b) = a ∪ Ø = a   and   a   is non-empty by definition.

Example. The Power Set of any set A is not a partition of A . Theorem. The Power Set of an EP set does not exist.
Proof for a two element set A = {a,b} : in {Ø, a, b, {a,b}} the first element is empty and the last element is not disjoint with the middle {a,b} .
Non-existence of the Power Set wipes away the Cardinality of the Reals and the Continuum Hypothesis, when founded upon EP sets eventually.

Example. Finite Ordinals are defined in standard set theory by putting more and more curly brackets around the empty set,
like in: 1 = {Ø} , 2 = {Ø,{Ø}} , 3 = {Ø,{Ø},{Ø,{Ø}}} , ...   But none of these sets is a partition of any set.
For example with 3 = {Ø,{Ø},{Ø,{Ø}}} : the first element is empty, and the second and the third element have the member Ø in common.
Thus Finite Ordinals can not be defined in EP set theory in this manner.

We are ready for another Theorem in the elementary partitons theory:   ∀ a,A : (a ∈ A) ⇒ (a ⊆ A)
Proof: a ∩ A = a ∩ (a ∪ x ∪ y ∪ z ...) = (a ∩ a) ∪ (a ∩ x) ∪ (a ∩ y) ∪ (a ∩ z) ... = a ∪ Ø ∪ Ø ∪ Ø ... = a .
Which by definition is the same as:   a ⊆ A  .

Theorem (ZFC Killer Axiom) :   ∀ x : ( x ≠ Ø ) ⇒ ( x = { x } )
Proof:  x  is non-empty, does not intersect other parts, fills up the whole of  x  .
Conclusion. The axiom that kills ZFC doesn't kill some other set theory.
Therefore EP set theory is not even consistent with a finitary part of common set theory.

Example. Consider the EP set  { a , b , c } . Then   { a , b , c } = { { a , b } , c } = { a , { b , c } } = { { a , c } , b } .
Proof: by definition  a , b , c  are all disjoint and non-empty.
Therefore the part  { a , b }  is disjoint from  c , the part  { b , c }  is disjoint from  a , the part  { a , c }  is disjoint from  b .
All these parts are non-empty and fill up the whole EP set.
Remark. This example clearly shows the difference between common set theory and Elementary Partitions set theory concerning membership.

Essential for the theory of sets is the distinction between set and element. Never mix up ⊆ and ∈ , a and { a } .
(S.T.M Ackermans / J.H. van Lint, 'Algebra en Analyse', Academic Science, Den Haag, 1976)
But .. an object is not uniquely defined by its members. There are many ways to partition an object into elements and define it to be an EP set.
The stone which the builders rejected, the same is become the head of the corner (Matthew 21:42).

Boolean Algebra and all of its laws are valid for EP sets, provided that all of the members in the universe of these EP sets are either disjoint or equal.
Then (and only then) we can form  { a , b } ∪ { a , c } = { a , b , c }  or  { a , b , c , d } ∩ { p , a , c , q } = { a , c } . And safely apply all the laws of Boolean algebra. No miracle, because EP set theory is essentially the same as Constructive Solid Geometry, which is the geometric (counter)part of common set theory.

The Physics of Infinity

An interesting question is if actual infinities could arise in the physical world. Actually, they do in our physical theories. Quantum field theory has to deal with infinite numbers and "renormalize". Cosmology is even more abundant of infinities. How about
Black Holes ? And the Great Singularity, which is assumed to be at the Origin of the whole Universe ? Whew ! But serious now: "... dass ein Physiker überrasht sein sollte, wenn er Phänomene in der Natur vorfände, in deren Beschreibung das Wort Unendlich nicht durch das Wort sehr gross ersetzt werden dürfte." (: Carl Friedrich von Weiszäcker).
Of course, there is no physical evidence that infinities actually do exist. Actually, there is physical evidence that actual infinities do not exist. I have done my best to collect some substantial, physical, empirical arguments. Most of these are certainly not my own. (Remember !)
  1. If the universe was infinite in space and time, then it would'nt become dark at night. The whole sky would be as bright as the sun itself. There would be an infinite number of stars, and all their light would have reached us, since there would be an infinite time for it to travel. This phenomenon is known as Olbers' Paradox . [ Google("Olbers' Paradox") gives more than 19,000 references ]
  2. What do you think of the ever increasing entropy in an infinite universe ?
  3. Infinite vector spaces are used in Quantum Mechanics. However, they all have a complete base, as if they were only vectorspaces with a very large dimension. In fact, these Hilbert spaces cannot be distinguished from a large, but finite vectorspace, as every physicist knows from experience.
Therefore, from a physical point of view, the Universe seems to be finite. Big Bang, Creation or Whatsoever ... And if it is finite in space, then it must also be finite in time, and vice versa: the Theory of Relativity leaves us no other choice.

So far about the infinitely large things. How about the infinitely small ?

  1. Fluid flow is described by a set of coupled partial differential equations. However, it is very clear that this must be considered as a mere illusion. The "differential" volumes are not really infinitely small. Far from that ! They should contain a lot of molecules, for the approximation to be valid. There is a beautiful text about this (due to Perrin) in Benoit Mandelbrot's 'The Fractal Geometry of Nature', first few pages (6,7).
  2. Space and time seem to be the only things that remain infinitely divisible. But there are strong counter arguments ! Quoting without permission from Landau and Lifshitz 'Quantum Electrodynamics', Introduction, page 2+3:
    > In the rest frame of the electron, the least possible error in the measure-
    > ment of its coordinates is:
    >                               dq = h/mc.  
    Here q=position, h=Planck's constant, m=electron mass, c=velocity of light. If you try to measure the position of an electron, you must send for example a photon to it. The electron is disturbed by this photon, which is known as the Compton effect. As a consequence of this, there is a definite bound on the accuracy with which you can locate the electron. This is expressed by the above formula. Any attempt to locate the electron within this interval is "severely punished" by:
    > ..... the (in general) inevitable production of electron-positron pairs in
    > the process of measuring the coordinates of an electron. This formation of
    > new particles in a way which cannot be detected by the process itself
    > renders meaningless the measurement of the electron coordinates.
  3. There is another (exellent) book, written by Léon Brillouin. It is called 'Relativity Reexamined', Academic Press (1970). Quoting without permission:
    > Einstein's clocks were supposed to emit extremely short signals and to
    > measure accurately time intervals between signals emitted and received.
    > In a word, an Einstein clock was a radar system, and its requirements
    > were thus very different from those of a frequency standard. ... 
    The modern atomic clocks are referred to by the above "standard".
    It can be shown by a simple thought-experiment that, if one tries to measure an electron by such an Einstein radar pulse, this pulse allways must be much wider than the following:
                       dt = h/mc^2        (t=time) 
    It thus is definitely impossible to measure (electron) time more accurately than this amount, due to the same Compton effect as in (2).
  4. There are some pathological theories in physics where infinities actually do arise, like Quantum Electro Dynamics (QED). In these theories, notions like "point" charges, with infinitely small spatial dimensions, are used explicitly. But what happens there is not a proof that "Actual infinities do exist". Instead this should be looked upon as a SEVERE WARNING:
    Absolute mathematical concepts like "point", "continuity" and "Identity" (above !) are not void of danger if they become an integral part of a difficult physical theory. Then the foundations of mathematics suddenly change into a piece of real world physics. Once our mathematical axioms are absorbed by a physical theory, then they become, in fact, assumptions about nature.
I don't think all mathematicians did design their basic concepts for such a purpose. Therefore, in my not so humble opinion, physics should underpin mathematics, in a certain sense. See the good (but old) signature:
* Han de Bruijn; Software Developer "A little bit of Physics would be   (===)
* SSC-ICT-3xO ; Landbergstraat 15,   NO Idleness in Mathematics" (HdB) @-O^O-@
* 2628 CE Delft, The Netherlands.    E-mail: J.G.M.deBruijn@TUDelft.NL  #/_\#
*   Tel: +31 15 27 82751.       ###

Sad Remark. When I say that Mathematics should be founded on Physics, then I mean "common" physics. I am not quite reluctant to accept "advanced" physics - like elementary particle physics or cosmology - as truly scientific. I'm a bit ashamed that these nowadays areas of interest seem to belong to physics anyway.

QED suffers from its infamous divergencies ever since it came into existence. People like Richard P. Feynman have devised some dirty "renormalization" tricks, in order to cure the very worst symptoms of this "infinity" disease. QED nowadays gives some of the desired answers. But, as Feynman himself points out in the famous "Feynman's Lectures on Physics" (part II), the difficulties already showed up in classical electromagnetic theory. The self-energy of point charges is infinite. Electrons cannot even move, if "actual infinities do exist" !

Physical theories can - and will - suffer from an inadequate foundation of their mathematics, I think. People should have the courage to postulate the following as their working hypothesis.

      Basic axiom: EVERYTHING IS FINITE
Challenge to everyone:
Give me one valid counter-example: a piece of evidence that the Infinite is actually useful in physics, in a theoretical or practical sense (independent of wishful mathematical thinking, of course).


Oh no, the spectrum of hydrogen is not a good example. Because the presence of other (hydrogen) atoms in the universe will blur the spectral lines, for high values of the numbers (m and) n in the Balmer formula:
                1/λ = R(1/m^2-1/n^2)   
More details are found in this article from the 1989 sci.physics thread.
As an example how infinities come into "existence" with a common physical theory, here is the well known equation of state for an ideal gas:
       p.V = n.R.T
Consequently, the theory of an ideal gas also predicts infinities. As soon as its volume V approaches zero (at a given room temperature T ) then the pressure p will raise to infinity. Every physicist knows, however, that such is not the case. For a manyfold of reasons. The most important being that a real gas does not exactly behave according to the laws for an ideal gas, certainly not for high values of the pressure. Instead of this, it will be subject to a change of state: it will become a fluid in the first place. And it even may become a solid, if pressure continues to squeeze it to still lower values of its volume.
If general lessons are to be learned from ideal gas behaviour, it could be something like the following:

The infinities associated with the ideal gas law are due to idealization of the real gas behaviour. They disappear if the natural gas is modelled more accurately. That is: as soon as mathematical models become more realistic.
If infinities are likely to occur within the realm of certain physical laws, then the matter subject to these laws will change state, in such a way that infinities are avoided. This effectively means that such "laws" will be no longer valid.

So far so good. Now think about those poor Black Holes that are subject to the Laws of General Relativity . . .  ;-)

Balls in a Vase

The following question is copied from a
thread in sci.math.
Suppose you have a giant vase and a bunch of ping pong balls with an 
integer written on each one, e.g. just like the lottery, so the balls 
are numbered 1, 2, 3, ... and so on. At one minute to noon you put 
balls 1 to 10 in the vase and take out number 1. At half a minute to 
noon you put balls 11 - 20 in the vase and take out number 2. At one 
quarter minute to noon you put balls 21 - 30 in the vase and take out 
number 3. Continue in this fashion. Obviously this is physically 
impossible, but you get the idea. Now the question is this: At noon, 
how many ping pong balls are in the vase? 
And here come some answers, according to Mainstream Mathematics. But there are quite a few opponents to this mainstream view. Here are some: Then the thread becomes monopolized by a character called Tony Orlow. And starts to grow to infinity, indeed ! :-(
So we shall stop here. David R Tribble summarizes very well the result so far:
Well, first of all, there is no consensus in this thread; there are 
three camps: one camp says that there will be zero balls in the 
vase at noon, another camp says that there will be an infinite 
number of balls in the vase at noon, and the third camp says the 
answer is indeterminate.
OK. Time to make up our mind. Who is right and who is wrong ?

The good news is that there is no sensible disagreement between all of the debaters about the timestamps between 1 minute before noon and just before noon.
We can say that the number of balls (picture) Bk at step k = 1,2,3,4, ... is:
Bk = 9 + 9 . ln( -1/tk ) / ln(2)   where tk = - 1/2k-1 for all k ∈ N .

But, within this formula, there is no timestamp at noon. Because this would involve k = ∞ , which is nonsense in any respect.
I have marked the following poster by rennie nelson as exceptionally right to the point:

[ ... ] . There is no noon in the problem as written, every event
happens before noon. The universe of the problem is not the same as
the universe of the question. No amount of reasoning or deduction will
overcome this. 
The bad news is that mainstream mathematics nevertheless claims to have a solution at noon and says that the vase must be empty then.
Wherefore by their fruits ye shall know them (Matthew 7:20). But, sigh! Why not stick to the solution where we all can agree about?
And admit that any "solution at noon" is ... beyond the scope of mathematics, ... as a science ?

Political Economy of Sets

Virtually any kind of "Modern" Math is based upon Set Theory, despite the fact that set theory suffered from (Russell's) paradoxes from the very beginning. Any other mathematical theory would have been assassinated by such obvious and serious shortcomings. It is quite remarkable that Set Theory has survived in the first place. Big surprise; it has even become the foundation "par exellance" whereupon Modern Mathematics is based !

From a rational point of view, this must sound like a true miracle. It strongly reminds of the way all kind of superstition beliefs are still going strong, despite their manifest lack of scientific content. So there must be something out there which is even more convincing than logic. Before going into details, let me tell you that a satisfactory explanation - according to my not so humble opinion - can be found only iff people dare to recognize that Mathematics, too, is just a humble activity of human beings. There is nothing heavenly about mathematics. At least, it is no more divine than for example composing music or writing that godly book. This implies that mathematics is bound to historical and social restrictions, in the very first place. Yes, I want you to get rid of the idea that "Mathematics is independent of society" !

Something out there must be more convincing than logic. Let's see what it is. How can somebody conceive the idea that the whole of mathematics is made up from nothing else but Sets ? This has only become possible because society itself has adopted the shape of an "ungeheure Warensammlung" (unprecedented collection of goods: Karl Marx in 'Das Kapital'). Is it a mere coincidence that the birth of Set Theory has its social analogue in the enormous accumulation of all kinds of goods, which marks the turn of the century (1900) ? Is it a mere coincidence that Georg Cantor's father himself was a merchant ? So his son became very familiar with those huge "sets" in the storehouses of his family.

So we may conclude in the first place that the birth of Set Theory was inspired by social circumstances. But this is not the end of the story. Even nowadays, nobody can think of an idea which better fits the view of the Capitalist System on society. Go to a supermarket, and convince yourself ! It all means that rational arguments are not good enough, in order to deprive Set Theory from its predominant role in mathematics. Read my lips: I don't want to get rid of Set Theory as a whole, I only want to deprive it from its predominant position.

Mathematical concepts originate and become important within the context of our human society, with all its non-logic and non-scientific taboos. But it is also thinkable that certain concepts will not originate in the given social circumstances, simply because such new concepts would have unacceptable economical and political (and personal) consequences. New ideas will not come, essentially because, deep in our heart, we don't want them. So yes, further progress in mathematics may be inhibited by the way our Western "civilization" is organised, as an ungeheure Warensammlung and nothing else matters.

We all watched, but we saw nothing, felt nothing but fear and the fierce wanting not to become like them.
(: Dante's Divina Commedia)

Why am I doing this ? Because I don't believe that people with wrong thoughts can nevertheless do right things. I'm well aware of the fact that thoughts of a Mathematical Nature form only a negligible part of human thinking, these days. Most of the time, our level of sophistication does not exceed the needs of everyday Greed. Not anymore. And most of our so-called "creativity" has been swallowed by Commercial Interests altogether. Yes, the world has become a Giant Marketplace. Run the risk of being a subject of disdain, if you dare to attach value to something else than Marketing and Management. Why to be a Scientist, when you can be his Boss, huh ? But, with the commercialization of everything, telling lies has become the standard way of communicating which each other. Don't be silly ! Especially standard Set Theory is reflecting quite accurately the peculiarities of Capitalist Society. What else to expect, if all the important decisions are made by the Money that we Have, and not by the Humans that we Are !

Function = Mapping + Time

The classical definition of a function sounds like this.
A mapping, or a function, F from a set A to a set B is a subset of the Cartesian product: F A × B .
A function is thus a kind of relationship between A and B , which has the following property, in addition:
          ∀ (a ∈ A) ∃! (b ∈ B) : (a,b) ∈ F
With each a element of A there exists exactly one b element of B , in such a way that (a,b) is an element of F.
What could be wrong with such a definition ? Let's devise a simple example, the function   y = x2 when defined at the natural numbers (: A = N). Then form the set B, as follows:

A12345 6789101112 131415161718 192021222324 25  . . . . . 
B1491625 36496481100121 144169196225256289 324361400441484529 576625  . . . . . 

The venom is in the tail. Because, as soon as these theorists think that there is sufficient evidence, then I would shout: Give me more, give me more ! And, to satisfy my needs, they will busy again, for some  . . .  time. Now that's the secret ! What's lacking in the classical definition of a function is simply the fact that it costs a finite amount of time to make the calculations, I repeat: the calculations, that are needed to create the set B, with the desired numbers in it. The classical definition, on the contrary, is completely static. Not a single indication of the work, the labour, the physical effort, the dynamics of building the products in set B from the commodities in set A . Instead, the production costs are simply denied. The arduous creation of value is reduced to nothing but a timeless relationship, that simply exists. Labour is reduced to just a heap of commodities and a heap of products, without the time-consuming process in between, without the dirty smell of sweat. Considered in this manner, the classical function definition does suit well to the pipe dream of some of our enterpreneurs.

It shouldn't be surprising when such a careless and biased abstraction, sooner or later, will prove its vulnerability, in both theoretical and practical applications. But instead of considering a reform of the classical function definition, which would have been relatively simple, these theorists went on and they invented a whole bunch of "new" concepts. To mention a few: function (of course), (differential) operator, mapping, dynamic system, Turing-machine, (Markov) algorithm, lambda calculus, (Post) productions. Each concept being meant to encompass still other aspects of the same process of getting the job done. With the advent of digital computers, this bunch of function definitions has been extended once more, with nomers and mis-nomers like: subroutine, procedure, program, script, executable, command, the methods of a class. All too many names for too many, rather trivial variations on one and the same theme. Quoting a favorite mathematician, without permission: There is perhaps no better example [ ... ] of missed opportunities than in the treatment of functions.