Consider the concept: "The most important insight obtainable from reading 'The Adventures of Huckleberry Finn' by Mark Twain". Although this means something to many people it is quite obvious that it cannot be defined, since this word implies a certain amount of precision and uniqueness. So most concepts that depend on a receiving individual will miss this precision due to the variability in the individuals and their reactions.

On the other hand, the lack of definitions in such situation is not dramatic and hardly comes as a surprise to unbiased people (taking into account that only a minority of people know what a definition is). So, one probably has to narrow down the kind of concept one has in mind to give the question a non-trivial character.

Dear Ulrich,

Your post is very very clever, but either I have not understood it accurately or you have defined the concept. Your definition is "The most important insight obtainable from reading 'The Adventures of Huckleberry Finn' by Mark Twain".

Notice that my question is not whether or not there is a concept the definition of which cannot be understood accurately. By contrast, the core of my question is the definition existence, instead of its understandability. In fact proving non-existence of any entity is always a difficult task. In general, non-existence proofs are based upon some contradiction derived from existence assumptions. In other words, if it is possible to prove that the assumption that every concept can be defined leads to a contradiction, then the question is solved.

Do you mean spoken languages or also formal languages?

It does not matter. Any language that a human can handle.

Any sign exists within a language. A concept outside a language does not exist. I am not restricting language to human language.

Then, if you mean 'physically' (in Nature) meaningful concepts, I would expect the answer to be: No, there are no such concepts .

Dear Louis,

I cannot understand your claim. Concepts I know are inside my mind. Sometimes formalized by means of some language and sometimes by means of intuition or perception.

When I feel something, say a toothache, I feel it in a language-free way. Nevertheless, I can define this sensation in terms of any natural language I know.

Indeed, anybody seeing my toothache definition, perhaps does not interpret its meaning as I do; but my question does not require that every defintion can be understood anywhere by anybody. In any case, concepts are equivalence classes. For instance, the concept of book involves a lot of different objects not only those make from paper, but also ebooks consisting of bit sequences.

I forgot to ask you what you mean by "concept".

Dear Juan-Esteban.

Well, "concepts are equivalence classes". If you apply just this statement to my literature example you will see the problem with its definition, that I tried to point out. Can you give any second member in the equivalence class 'defined' by my example?

I guess you will conclude that the concept is 'not defined clear enough' that such a search makes sense.

What about undecidable languages (i.e, I speak of those in Computability Theory)? We know the concept of a decision. There surely exist formal languages which are undecidable, but there necessarily exist infinitely many undecidable problems. Can we describe a formal language that can conceptualize any undecidable language from a decidable language? No. Why? That's a consequence of the Halting Problem. There's one for you. Though I would think it is really stretching it as it is a "meta-meta" result, but it is still valid.

Dear Juan-Esteban,

When you feel a toothache, your feeling communicate to you in the feeling language that something is wrong with your theet and you do not use it to make things worst, etc. This natural language is quite efficient at making you understand what is going on and what to do. In the living that are good reasons to believe that all the processes are regulated into all kind of language networks. This is the biosemiotics ways to see biology. So if you have a concept in your mind that you have not yet express explicitely into ordinary language or mathematical language, there are good chance that it exists already into another lower level language network of the mind, the language where creative work actually take place and which ordinary language is in intimate connection otherwise nothing would make intuitive sense. It is what I was meaning by my previous post.

Dear Louis,

Do you mean that a baby or an animal either knows some language or he cannot feel any sensation?

Dear all,

Let E denote the set of all concepts or objects that can be defined by means of some language. Indeed, we have just defined E; therefore E belongs to E; consequently E is self-contained. Likewise, E contains every set that can be defined. Now, let us consider the statement,

(1) ∃ x: x ∉ E

If the statement (1) is false, then, for every set x, the relation x ∈ E holds. In other words, E is the set of all sets, which leads to the well-known contradiction called Russell’s Paradox. Thus, the statement (1) must be true.

Finally, if (1) is true, there is at least a set x which does not belong to E; hence, as a consequence of the definition of E, such a set x cannot be defined.

Juan-Esteban, very good.

Russel's paradox parallels Gödel's incompleteness theorem, which underprops a general reality - to wit, sets attempting to describe wide swathes of external reality tend to be open or uncircumscribable

Juan-Esteban,

x is first defined as object or concept and then after (1) you are talking about set x.

Statement (1) is true if there exists a concept that cannot be defined. In mathematics, existence pre-suppose a possibility of definition. That is not true in life but math is less general than life.

Dear Louis,

I appreciate your clever arguments. With your contributions the thread is keeping alive. You have said that existence of an object requires the possibility of defining it. I think that this is not an axiom of any mathematical theory. It is possible to define collections without defining every of their members. Even the existence of some entities can be proved ignoring their nature and definition. Likewise you can define objects that cannot exist. In any case, my claim does not contradict your "axiom". See why.

My question does not deal with general definitions, but definitions “by means of some language”. Languages consist of sequences of symbols lying in some finite alphabet. Accordingly, every definition consists of a finite sequence of symbols. You can also consider countable symbol sequences. What about non-countable alphabets and infinite definitions?

Consider that languages are open constructions. For instance I can create a new word in English, for example the word “aletheion” and I can define it by means of existing English words together with pictures, images or actions. In fact, languages are increasing as we are creating new theories.

Let A_0 be a finite set of words in some language. Now, you can append new words obtaining an extension A_1⊃ A_0. Iterating the process you can obtain an endless sequence of language extensions A_0 ⊂ A_1 ⊂ A_2 …. In general, for every couple positive integers n < m, there are definitions in A_m that cannot be denoted by means of members of A_n. If the extension sequence is infinite, then, at each step, there are always objects or concepts that cannot be defined. You can avoid this inconvenient using an infinite language A being the infinite union of all extensions. By means of A every object can be defined. Notice that if there is a definition in A for every set, then A is equipollent to the set of all sets. If you accept the existence of such a language, then you are in the scope of Russell’s paradox.

Summarizing, you can accept that for every concept O there are at least one language by means of which O can be defined, but the assumption of the existence of a language having the capability of defining all objects leads to the contradiction lying in the Russell’s paradox. Consequently, such a language cannot exist, because its existence leads to a contradiction. As a consequence, languages must be open constructions.

After Russell's paradox every set must be open, that is to say, must be extendable by adding new members. Accordingly, it does not matter how large is a definition set, there are always new definitions which can be added. Neither object sets nor definition sets can be completed. In addition, every consistent system must be also incomplete (see Gödel theorem).

There have to be concepts that are not known to us yet. Such concepts may actually need new terminologies for explanation. Till then, we shall have to wait.

For example, to explain the gravitational matters, it required a Newton to come into existence to discover calculus. Before that, those things were unexplainable because the mathematical language to explain the matters was not there.

Dear Hemanta,

You are right. Even real numbers cannot be defined before introducing the concept of limit. Likewise, If your sandwich has been stolen, you know the "existence" of a thief, but you cannot define him unless you know if the thief is a human, a cat or a monkey etc. Thus, existence can be proved before knowing the corresponding definition.

Dear Juan-Esteban ,

There is not only one type of logic and there are probably different ways to build new mathematics without sets and without any Russel`s paradox so as to express all kind of new type of concepts. It is strange that mathematics which is presented as the language of the timeless platonic world keeps changing and is so dynamics.

Dear Louis,

What I have said is that Maths, like mathematical language are increasing, but are not rejecting assumed constructions. Increasing means progress.

In any case, maths are not philosophy.

Finally, I have voted up your post, because I agree with you. There is not only one type of logic. In addition, with multivalued logic every contradiction can be avoided.

In fact I prefer to live in the kingdom of multivalued logic, but, nowadays is not an easy task. See the following quotation from M. Planck.

"An important scientific innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning."

Max Planck (1858 - 1947)

Dear Louis,

You have claimed that

"There is not only one type of logic and there are probably different ways to build new mathematics without sets and without any Russel`s paradox so as to express all kind of new type of concepts"

I agree with you, besides, I think that it is worth opening a new thread dealing with this topic.

very good question... this is keeping me occupied

Dear all,

After Louise Brassard's anwser, I have posted a new related question:

>> Is the multivalued-logic the best option? >>

"Is it possible to know whether or not there are some concepts which cannot be defined by means of any language?"

It is a very strange question and I believe that Louis Brassard has pointed out that the positive reply involves a contradiction: "some concept which cannot be defined by means of any language" is a definition of concept in a language, therefore the reply must be "no", because the positive reply involves a contradiction. No need of the sophisticated Gödel theorem.

By the way, Juan-Esteban, you disagree that existence of an object requires the possibility of defining it and you wrote that this is not an axiom of any mathematical theory.

From an intuitionistic point of view, you are wrong. Descartes said, in French, "c'est du connaître à l'être que la conséquence est bonne". In English, Descartes meant that before wondering if a thing exists, you have to wonder about its definition. From a formalist point of view, coherence is enough to define any mathematical object. But note that this formula $\exists x (\neg Fx \lor Fx) $ holds that exists one x that is either F or never F. It is perfectly consistent, because the negation of this formula is absurd. But can you seriously hold that it defines the existence of an object? I do not believe that it is a good option to get a clear idea of what "existence" means. From an intuitionistic point of view, it is an "axiom" that $\exists x Fx$ is true only if you can show in your mathematical theory some object as witness of this formula.

Dear Joseph,

My viewpoint is mathematical, while you go through a philosophical path. Mathematical proofs must be based upon some axioms.

In any case, I think that you are confusing knowledge and existence-inference. Notice, that Louis Brassard has discerned the difference and to reject my claim invokes a new logic.

Consider the following questions.

A baby can know his mother before being able of handling any language and defining the mother notion. In fact human thought is based upon intuitions. Definitions take place in a higher abstraction level. Since you use a philosophical viewpoint, I try to prove it by this method.

¿What a thing is a definition given in any language L? Indeed, it is a sequence of words of L. Suppose that w1 w2 …wn is the sequence of words in English defining an object O. According to your claim, the meaning of each of these words can be also defined. Suppose that ww1 ww2 … is the definition for the meaning of the word w1. By the same reason, ww1 can be defined by another sequence of words www1 www2…. and so on. This is an endless process that requires an endless set of definitions. Thus, according to your claim, the knowledge of one definition requires an infinite set of others definitions. Of course, there are objects that we know by intuition or perception and not by definition.

¿Do you want to know how?

Suppose that I mix three colors, for instance 0.4 red, 0.3 blue and 0.3 yellow and I show you the result and say that this is the color XX. You know the color XX, but you do not know its definition, that is, the mixture 0.4 red, 0.3 blue and 0.3.

You know the color XX by perception; however to define XX it is required knowing its nature and composition.

Likewise, when the police find a dead man, they know the existence of a “cause” of his death. Nevertheless, until they do not get at the death “cause” only the cause existence is known, but its nature and definition are ignored. This is the method I have handled. Negating the existence of objects that cannot be defined we find a contradiction (Russell’s paradox); consequently we can know the existence even ignoring their nature and definitions, at least, inside the kingdom of dichotomic logic and classical set theory. In any case if you do not agree, the proper method consists of pointing out a mistake in my proof. If you prove the converse statement using different methods, you are creating a new set theory, by no means you are rejecting my proof, you are rejecting the classical set theory. It is a very ambitious task...In fact, there are several set theories, but my inference goes under the rules or axioms of the Cantor-Goedel's one. In any case, I am not a Cantor believer, for instance you can see my paper

https://www.researchgate.net/publication/241698078_The_Existence_of_Intrinsic_Set_Properties_Implies_Cantor's_Theorem._The_concept_of_Cardinal_Revisited

Article The Existence of Intrinsic Set Properties Implies Cantor's T...

Dear Juan-Esteban,

Yes, I am more and more inclined to believe that to take classical logic and classical mathematics as the Core Logic of human knowledge is a philosophical and a scientifical mistake just because Boolean algebra is a particular case of Heyting algebra and because intuitionistic mathematics are in fact richer than classical ones.

It is wrong that there is one and only one right logical-mathematical viewpoint to base a correct philosophical system. I believed for example that the Quinean option was strong enough, but I have realized that it is based on misunderstandings of what intuitionistic logic really says.

I do not know if I am confusing "knowledge and existence inference", maybe. But for me, knowledge is not necessarily the knowledge of a being that exists (for example I know that p => p is always true, but by knowing this analytic truth, I do not know one existence). And I make a distinction between existence, and existence-inference.

Please, explain to me why do you believe that I'm making here the confusion that you mention.

Thanks for your paper !

Dear Juan-Esteban,

You wrote : " Negating the existence of objects that cannot be defined we find a contradiction (Russell’s paradox); consequently we can know the existence even ignoring their nature and definitions, at least, inside the kingdom of dichotomic logic and classical set theory. In any case if you do not agree, the proper method consists of pointing out a mistake in my proof. "

In my opinion your interpretation of what Russell paradox says is too large or not precise enough. Russell's paradox says only that the set of all sets that are not elements of themselves is a contradictory set. Being a contradictory set, this set cannot exist (except if you believe Graham Priest). So if you proof is based on Russell's paradox to hold that we can know the existence of a thing, even ignoring its nature and definition, that seems to me to be a first mistake.

Best wishes,

Jo.

Dear Vidal-Rosset

You have pointed out that “Russell's paradox says only that the set of all sets that are not elements of themselves is a contradictory set." It is not this paradox what I am dealing with, but this one http://en.wikipedia.org/wiki/Universal_set. This is to say, the paradoxical concept of the set of all sets. This paradox leads to the Russell’s one. Indeed if the set of all sets exists, then it contains the set of all sets that are not elements of themselves, because this collection is nonempty. For instance, the set of all books is not element of itself.

This is why, in set theory, the set of all sets is not considered a set, but a class. See the first chapter of http://katmat.math.uni-bremen.de/acc/acc.pdf

That is to say, the set of all sets cannot exist, but the “class” of all sets does exist.

I think that you have not read my previous messages. In one of them I have seen the possibility that for every object there is a language by means of which can be defined; however a universal (total) language defining every object is not extensible, therefore leads to the universal set paradox, which leads to the Russell’s one.

What classical theory negates is the existence of any universal collection that cannot be extended. Accordingly if you accept that there is the set D of all definitions, and there is nothing outside D, then D is not extensible and cannot exist. To avoid this kind of contradiction, any collection you can build must be a proper sub-collection of another one. Notice that, if every object is definable, then there is a one-to-one correspondence between the class of all definitions and the class of all objects.

In any case, have you taken into account that if there is a definition for every concept, there is also a definition for its definition and iterating the process an infinite set of definitions is necessary?

Best regards.

J. E.

P.S.

In fact maths deal with abstract constructions based upon axiom sets. It does not matter whether or not the resulting construction describes some part of Real World, that is, no model is needed.

Dear Michael,

It is not easy to explain the question without using some formal language together with mathematical concepts. Nevertheless, I will try to expose some ideas minimizing the use of maths.

Consider that the claim “every concept can be defined” denotes the possibility of stating a monogamist marriage between each pair consisting of a concept and a definition. Between the members of a collection consisting of 20 men and another consisting of 25 women, it is clear that not every women can be married. In other words, if this is the case, not every concept can be defined. From some viewpoint our question is answered by similar machinery. There is no limit to build concepts and objects. By contrast, any language consists of words of some finite alphabet. Every definition is nothing but a finite sequence of words. As a consequence the collection of definitions is limited by the underlying construction machinery in any language, while there is no limit for the construction of possible objects or concepts. As a consequence, it is no matter the used language, there will be "non-married" concepts.

I am aware that this explanation is not as accurate as mathematical method requires; however it is easier to understand.

Regards.

Recall another questions of mine: "Is there a finite definition for every real number? " In the corresponding thread it was proved that, as a consequence of Cantor’s theorem, there are real numbers that cannot be finitely defined, however they exist. In fact, an infinite definition, that is to say, an endless one, defines nothing. Thus, there is no definition for those real numbers that cannot be finitely defined.

To illustrate this topic, consider a real number 0.c1c2c3…. the figures of which are chosen at random. Since there is no predictable pattern in the figure sequence c1c2c3… in general, there is no finite definition.

I know that this illustration is not accurate, this is why I have termed it as illustration instead of proof. The proper proof lies in the referred thread.

It is worth noticing that the concept of language is bounded to human mind. Perhaps the reader is tempted to claim that some heavenly being can handle a language by means of which every concept can be defined. Perhaps it is a great truth, but RG is devoted to scientific research, instead of magic thought.

Mathematics comes from attempts at solving very pragmatics problems. So originally it was not separate from physics. I cannot see the usefullness for physics of numbers that cannot be defined. Maybe the concern to be usefull for physics is not important for most mathematicians. Joseph mentioned intuitive mahematics. In such framework, such undefinable numbers would not exist. The mathematical language to be usefull has to be close to reality in the sense that the informational content of the mathematical concept has to be finite allowing an embodiement in this world. I know it is leading us away from the question.

Dear Louis,

What is applicable in physics is called “applied mathematics”: however there is a lot of mathematics with no application, at least nowadays, I am not a fortune teller; are you?. Consider that poetry has no application; however one can enjoy reading poetry. There is a lot of art and poetry in non-applied mathematics. See some examples.

Every theorem in number theory.

The existence of perfect numbers.

The existence of amicable numbers.

The theory of Sierpinsky spaces.

Fermat last theorem.

Transfinite cardinals theory.

………

………

There are some people who are able to enjoy with poetry, imagination, beauty and genius lying in each of these great creations of human mind. It is clear that not every human can. There are also people who are not able to enjoy with Mozart's music, but I think this is a lack of these people, by no means a lack of Mozart’s genius.

Every time I have published a paper, a friend of mine asks me: how useful is your work? If my answer is: it is of no use, I have observed that he is happier. I like people to be happy, accordingly I prefer to say that the existence of concepts that cannot be defined is a useless thing.

See what a great mathematician have said:

A good mathematical joke is better, and better mathematics, than a dozen mediocre papers.

John E Littlewood (1885 - 1977)

Best regards.

Juan-Esteban

Dear Michael,

The marriage is an illustration, it is not a proof, in any case, consider the following facts.

A language L can be regarded as a machine to create definitions. Suppose that there is another machine O creating objects. If the machine L works slower than O, then there will be non-married objects. It does matter that there will be a husband for each non-married object in the future.

By contrast, you can suppose that when both machines will be finished their jobs, every object will be married. To be finished it is required the existence of actual infinity. This is not a universally accepted axiom. Consider this quotation.

>>Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.>>

H. Poincaré (1854-1912).

Nevertheless, Cantor's set theory, which is widely accepted, assumes the actual infinity existence. Likewise, Cantor's theorem shows the existence of non-countable sets. If you accept both actual infinity existence and Cantor's theorem, then you are assuming the existence of non-countable sets. However, the set of all definitions in any language is countable, because it is generated by a finite set (alphabet). Accordingly, if you assume Cantor's theory only a countable set of objects can be defined in any language.

As a consequence there are an uncountable set of objects which cannot be defined.

If you reject Cantor's theory and the actual infinity existence, then I have shown that the job of the machine L can be never completed.

Notice, that my proof is a mathematical construction, therefore it cannot be analyzed by means of philosophical methods, at least by a proper one.

Finally, science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house. Thus, claiming isolated facts is not a scientific method. Consider that I have inserted my proof inside the architecture of set theory.

I believe that Wittgenstein demolished Russell's Dilemma inThe Philsophical Investigations. All significant concepts, grundnormem, are undefinable in language. The key example is God. As Maimonides demonstrated so succinctly God is the reason for being but nothing meaningful can be said about God.

All your points are well taken.

I agree that field of creation such as mathematics has to be done along unconstraint personal aesthetic choices that should not have to be justified along short term, short sight pragmatic views. What is today useless might be tomorrow new foundation for physics. The last three revolutions in physics (Newtonian, General relativity, Quantum physics) took place after a mathematical tool kit was invented. The mathematical tool kit of Newtonian physics:

- Cartesian geometry : merging of algebra with Euclidean geometry

Geometry is become algebraic and vice versa

- Calculus and differential equation, initial conditions

The mathematical tool kit necessary for modern physics:

- Gauss differential geometry

- New general conception of geometry as group of symmetry

- Invention of Rieman geometry

- Reformulation of physical laws in terms of symmetry

- Invention of Hilbert space

What will be the mathematical tool kit for the new physics? The new physics is necessarily the physics of the origin of the cosmos. I suspect that since the core of this theory is a mathematical tool kit, the mathematical construction of this mathematical structure correspond to evolutionary step of cosmic creation. It is why I think that the mathematical construction has to be simple and not involve hyper complex structure such as real numbers.

Those are the speculations that were driving my last post.

Just as a liquid takes the shape of its container, reality conforms to the human mind and its instruments when they are analyzed by reason.

We see the physical reality through mathematics and takes the form that these impose.

In any case, science either is not dogmatic or is not science. We can believe what we want, but we cannot know anything that we do not receive information. What we call the origin of the universe can only mean the moment from which it is possible to obtain information. By no means can mean the time before which there is nothing. The relativistic mechanics is built on the information obtained by each observer (reference frame). Quantum mechanics denies the principle of causality. What is certain is that science will never admit any dogma. All laws of science are provisional. You can believe that Universe was created, was generated or is eternal, but no one of these options is knowledge. You can believe in some of these options, but you cannot know any of them unless you modify the meaning of the term "knowledge". If you modify the meaning of the word "knowledge" the question becomes simply a semantic problem.

Dear Juan-Esteban,

it is true that our scientific perception of physical reality is strongly influenced by the mathematics we have at our disposal. But one should not go so far as to think that it is determined by it. Schrödinger was led to partial differential equations, and Heisenberg was led to matrices (which he did not know from mathematics so far!) when thinking about how electrons move in an atom. So it is certainly not mathematics alone that 'imposes the form'.

Of course I agree with 'science either is not dogmatic or is not science'. However, there is a problem with this. Many people these days hold the view that science itself is a dogma and can't reclaim higher rights than, e.g. shamanism. What can we tell them? Without going into theories one hardly can say more than that only science allows to build electromotors, NMR diagnostic machines, spacecrafts, atomic power plants and bombs ... . This is not convincing to those who think one should do without these things.

To: Juan-Estaben Palomar

According to you, 'Science either is not dogmatic or is not science'. You mean, either science is not dogmatic, or science is not science.

I could not understand. What can be the meaning of the expression 'Science is not science'?

Dear Ulrich and Hemanta,

If I want to sell copper rings at a very high price, I can label them as gold. That is called scam. Similarly, if a scientist sells a dogma as science, then he is cheating. At least if the term science is given the same meaning as ancient Greeks and modern scientists.

The prestige that has gotten the word science, because of the progress that has given us, tempts many to use the label "science" to sell dogmas.

Science is just a method not a doctrine. Science is not built with "truths," but with hypotheses that must be verified experimentally or proven rationally. The dogmatic doctrines are built by the inverse method. One decides that something is true and tries to match reality with his truth. When it is not possible, one denies reality.

Remember the RG thread entitled "can science and religion unite?

I think it is a misuse of both terms. The following question must be stated before: "Can several religions unite?." No such thing ever happened, because every religion is dogmatic. However, dogmatic does not mean false. Only means untestable, therefore non-scientific.

Dear Juan-Esteban,

I agree with the letter of what you said but I cannot understand why you thought it was necessary to say it. Is it sometthing in what I was saying that seem to go against the principle of empirical verification?

Dear Louis,

Not at all. You are right; but my explanation has to be considered as a response to the Hemanta's sentence: "I cannot understand " science is not science."

Hemanta is right to demand more accuracy. The sentence is a contraction of the following one: "dogmatic science is not science in the strict sense." This is why I have considered necessary to explain it, because sometimes the term science is misused.

In fact the word science sanctifies every sentence containing it. In my opinion that is the cause of being misused.

I presented a paper this July at the Workshop on Logical Quantum Structures at UNILOG2013, Rio which suggests the possibility of such concepts:

The paper:

http://alixcomsi.com/42_Resolving_EPR_Update.pdf

Presentation:

http://alixcomsi.com/42_Resolving_EPR_UNILOG_2013_Presentation.pdf

Regards,

Bhup

http://alixcomsi.com/42_Resolving_EPR_Update.pdf

Donald Goldstein wrote : "I believe that Wittgenstein demolished Russell's Dilemma inThe Philsophical Investigations. All significant concepts, grundnormem, are undefinable in language. The key example is God. As Maimonides demonstrated so succinctly God is the reason for being but nothing meaningful can be said about God."

There is at least one correction to add to this Wittgensteinian platitude: there are in some language L1 some undefinable concepts which are in turn definable in one language L2, and so on. But it is well known that Wittgenstein, at least in his Tractatus, did not accept that philosophy can be based on the acceptance of metalanguages. There is no semantic ascent from his point of view, and this point of view sounds therefore very odd.

@ Juan-Esteban : The belief to the existence of some absolutely indefinable concepts cannot be based on Gödel's first theorem of incompleteness. What this theorem proves is, that for any language in which Peano arithmetic is definable, one can define an undecidable sentence G. In other words, the logical form of this theorem is :

$ \forall x \exists y Gxy $ and in FOL one can prove that this formula *does not* imply this one : $ \exists y \forall x Gxy $ .

It means that there is no possibility to infer from Gödel's incompleteness theorem that there are some concepts which cannot be defined by means of any language. Note that the logical form of Russell's paradox is on the contrary : $ \exists y \forall x ..." but, again, this paradox leads to give up the claim of the existence of such a set.

You reply that these concepts are concepts of class, not concepts of sets, by reference to Von Neumann-Bernays theory. That is a good point, indeed, to make such a difference. But it does not allow to reply positively to the question of this thread: if you hold that *there is a concept of class* which cannot be defined by means of any language you are lead to define such a concept of class like a concept of set, and you find again the destructive Russell's paradox, or you are defining only the possibility of indefinite semantic ascent by your definition. But the poential infinite is a perfectly well definable concept, therefore, the reply to the question must remain "no".

Platonism in mathematics is a perfectly respectable philosophical position. But finally it is based on faith, not on proofs, and because truth transcends proofs in this philosophical system, it is a consistent position.

Dear Joseph,

You have written

@ Juan-Esteban : The belief to the existence of some absolutely indefinable concepts cannot be based on Gödel's first theorem of incompleteness.

I cannot see any relationship between any of my contributions and your claim.

I have mentioned Gödel first theorem to illustrate my claim. You must not confuse illustration with any proof.

I have proved the existence of concepts that cannot be defined by means of Cantor's theorem. In addition, I believe nothing. My claim is not a belief, but a theorem based upon the well-known Cantor’s one. If my claim is wrong, then Cantor’s theorem must be also false. By contrast, if Cantor's theorem is wrong, the existence of these concepts can be false, but my proof keeps fine.

It is very difficult to expose mathematical concepts in a philosophical scope accurately. In any case, I try to explain you what is the matter

The algebraic structure of every language corresponds to the categorical constructions called monad and partial monoid. Do not confuse with the Leibnitz monad notion. A monoid is generated by a collection of elementary pieces called generating system. There is no finite generating system that can give rise to all possible monoids. As a consequence to define any object, a language must be generated by an infinite set of words. Who knows categorical algebra can see this topic easily. Nevertheless, I try to explain it via a simile.

Consider a language X from which you do not know any word. If you have a dictionary D of X, then you have the definition of each word. Can you get at the meaning of every word of X by handling the dictionary D? Indeed, you cannot. If you are looking for the meaning of a word W, the corresponding definition is stated through other words of X. The definition of each of these words is also described by means of other words of X and so on. To understand every definition in D you must know a minimal set G of words in X. The required minimal set G is what is called "generating system" of the partial monoid X.

Now, suppose that X is an infinite language. Since the generating system G is generated by a finite alphabet, then X is countable, that is to say is equipollent to the natural number set N. However, Cantor’s theorem shows that the collection of all subsets of any infinite set is non-countable. As a consequence there is also a non-countable set of objects that cannot be defined, because every language maps definitions into the corresponding defined objects and, according to Cantor’s theorem, this map cannot be surjective.

This deduction is so simple and elementary that the existence of a language generated by a finite alphabet by means of which every concept can be defined is nothing, but a counterexample for Cantor’s theorem. If you can prove this topic, you are guaranteed immortality. But I am speaking about mathematical proofs. Opinions, sensations and believes live outside mathematical architecture.

Since there are contributors refusing to accept the existence of concepts that cannot be defined, I think that a explicit proof must be posted.

Proof,

Let A denote the alphabet of any language L. Since A must be finite, there is a bijective map F from A into a finite set E = {1,2,3...n} of positive integers. Since every definition is a finite sequence of symbols in A, then there is an injective map F* from the set D of all definitions in L into N (natural number set). For instance, if p1, p2... is the sequence of al prime positive integers, for every defintion w1 w2 ...wm the assigned interger can be

F*(w1 w2...) = p1^(F(w1)).p2^(F(w2))....pm^(F(wm).

and by virtue of the Euclidean nature of the integer ring this map is injective. Thus, the map F* assigns a positive integer to each definable object. If the domain of F* contains the class of all possible objects, then there is an surjective map F* from the set of all possible definitions into the class of all objects. As a consequence the class of all objects is countable, which contradicts Cantor's theorem.

Summarizing, if Cantor's theorem is true, there is non-countable set of objects that cannot be defined by means of any language generated by a finite alphabet.

Dear Juan-Esteban,

I have found it useful to view Cantor's theorem as asserting that, given any countable set S of objects, there is no algorithm that can verify the claim that some set P(S) must contain all the possible subsets of S.

So any such set could, perhaps, be considered as a concept that can be denoted arbitrarily by a notation such as P(S) in a first order language L, but it cannot be defined in L.

Of course any language can circumvent this constraint by postulating that P(S) also denotes a valid object in every sound interpretation (model) of the language.

However, this only shifts the onus to that of establishing that the language does, indeed, have a sound interpretation.

By a sound interpretation I mean a model in which the assertions that are are assigned truth values by the rules of the language are also assigned corresponding truth values unambiguously by these rules under the interpretation.

This may not be an issue so long as I intend the language to only express my mental concepts (an artist's intent); I can always subjectively treat the mental constructs that I intend to express in the language as a sound interpretation of the language.

It is an issue if I intend the language to offer an unambiguous correspondence between my mental constructs and those of another intelligence (the critic's intent).

In that case---ideally---we both need to objectively identify a common interpretation of the language in which the assertions that are assigned truth values by the rules of the language are also assigned corresponding truth values unambiguously and finitarily by these rules under the interpretation.

Regards,

Bhup

Dear Bhupinder,

Cantor's theorem says nothing about algorithms, but bijection existence.

On the on hand, Cantor's theorem shows that "there is no bijection between a set S and its power set P(S)."

http://en.wikipedia.org/wiki/Cantor's_theorem

What I take from this fact is only the existence of uncountable sets.

On the other hand, every free monoid generated by a finite set (alphabet) is countable, as I have shown in my previous message by means of a Gödel-like numbering function F*.

These facts lead to the unexistence of any injective map from sentences of any language into any uncountable set. Accordingly, for every language, there are a lot of members of any uncountable set that cannot be defined.

If you do not agree, you must show what is wrong either in my proof or in Cantor's theorem. Any other philosophical claim constructed outside from the axiomatic method of maths does not fit into this topic.

Finally, it is clear that if an object O cannot be defined in a language L, then you can create another language L1 in which O can be defined. It does not matter, while the language is generated by a finite alphabet there is always an uncountable set of concepts that cannot be defined.

It is an easy exercise to prove that, if there is a finitely generated language by means of which every object can be defined, then Cantor's theorem is wrong. It is a very ambitious task. Who can prove this claim, deserves a chapter in the history of maths. I encourage the readers to do it.

Dea Juan-Esteban,

What is the meaning in your proof of the expression : " the class of all possible objects" ? What is a possible object if its possibility is not done by defined rules of construction? Is possibility just given by consistency? In my opinion consistency is just a condition sine qua non for possibility, a correct requirement from a logical point of view, but it is not enough from a mathematical point of view.

My point is the following one: if you accept in your proof the non-constructivity of non-denumerable, it is not at all surprising that you can prove the existence of an infinity of things, like real numbers of classical mathematics which cannot be all defined. But if you play another game, I mean if you say : by possible objects I mean such and such objects that I can define with such and such rules, precisely, it will be harder for you to prove the existence of concepts that cannot be defined, if existence of objects is defined via *definite* rules of construction, and not only, as we do in classical logic, via consistency only, i.e. the tricky business of formalist mathematicians.

Dear Joseph,

The meaning of the phrase "the class of all possible objects" can be deduced easily from the context. What I have shown is that, for every language L, every uncountable set contains members that cannot be defined by means of word sequences in L. Accordingly every uncountable mathematical construction satisfies this requirement.

Thus, real number set, vector spaces over real or complex number field, etc belong to the scope of my proof. Accordingly, each of these collections is a possible object.

Likewise, the union of these collections is the class of all possible objects that fit into my proof.

Remember that the proof is very simple and can be synthesized in the following terms.

1) The set of all phrases of any language generated by a finite alphabet is countable, as I have proved by means of a Gödel-like numbering function.

2) Cantor's theorem shows that there are uncountable sets.

3) As a consequence, there is no surjective map from the collection of all phrases generated by any language into any uncountable set.

The three statements above are the proof. Do you see something wrong in them?

Dear Juan-Esteban,

Wheras I'm at home with Joseph's thoughts, I have difficulties entering your mental world; can you help me? What can you say about 'object O' that is not defined. Don't you, by associating a symbol with the object, assume that it is a specific/particular, entity which it can't be without some kind of definition?

Dear Juan-Esteban,

I hinted earlier that the way you approach "concept"---in the light of our math. experience---cannot be relied on *for the purposes you want to address it*. Set theory and logic, as we know them today, are fundamentally inadequate for the clarification of "concept", unless you want to approach "concept" based on them, which is quite artificial and not interesting.

Dear Mutze,

Your are trying to start a Plato-like dialog. I have stated a mathematical proof of my claim. If you can show something wrong in it, I will appreciate your explanations. If you do not understand it, this is not my problem. I am not interested in opinions, illuminations or prejudices. I only pay attention to mathematical facts.

In any case, I can say you that about an object that cannot be defined you can know as few things as about any unknown object. Perhaps you know everything in the Universe, but it is not my case.

Best regards.

Dear Lev,

It does not matter. Where i have written concept you can read object. I have used both terms as synonymous. In fact every object is a mathematical construction and every mathematical construction is handled through a concept. Since in my proof only cardinality matters, I can use any of them.

Mathematicians use to do this kind of abstractions.

Dear Juan-Esteban,

I believe that I understand the meaning of your proof and also that it is perfectly possible to reduce it to ther domain of real numbers set because no language based "generated by a finite alphabet" can define every real numbers, there will be always an ifinite set of unspecifiable real numbers.

Okay.

But my claim is that this infinity of existent things has only as historical theoretical interest to make an harmony between the continuum of geometry and the number theory. It is a well known point of history of mathematics.

Nevertheless, it remains that matematicians working in theory of numbers are interested in knowing the properties and relations of defined numbers or properties and relations of defined class of numbers, and I am afraid that they find little interest to absolute unspecifiable numbers.

That's why, even if I am not a mathematician but only a poor philosopher, and amateur logician, I believe that intuitionistic mathematics are probably more interesting now, because we know that classical theorems of logic can be relativized intuitionistically as decisions on atoms. An interesting mathematical object is only a definable mathematical object, and procedures of decisions are finally more interesting than undecidability results, in my opinion.

Juan-Esteban,

"It does not matter. Where i have written concept you can read object. I have used both terms as synonymous."

You see, it just happened that the "object" and "concept" have occupied me practically all of my professional life. Again, although I was educated as a pure mathematician, it turned out that none of these can be approached adequately relying on the conventional math. language.

It's a long story, but basically, IMO to properly address them we need a radically new formal language, structural representation, which, in contrast to the point-based set-theoretic framework, requires a structural form of "object" representation.

Dear Lev,

I think you have misunderstood me. I will try to explain better. If you are trying to know whether of not two sets X and Y are equal in size, it does not matter the nature of their members. You can do abstraction of the member nature.

I suppose that if you are examining a pupil, you only pay attention on its knowledge and you do not take under consideration whether it is rich, poor, pretty or ugly. This working way does not mean that it does not matter to be rich or poor, but in examination purposes it does not.

Dear Joseph,

You have said,

"But my claim is that this infinity of existent things has only as historical theoretical interest to make an harmony between the continuum of geometry and the number theory. It is a well known point of history of mathematics. "

Pure maths deal with this kind of topics, If you are not interested in this topic nobody oblige you to contribute.

Fortunately, there is every kind of people. Some people that love music and some people that hate music. Some people that love pure maths and a lot of people that hate pure maths. Some people that love philosophy and some people that hate philosophy.

I do not care....I am so odd that I hate football. I am sure that football is a useless activity in order to describe our Universe. In any case, when I opened this thread I did not seek to save the world. Nevertheless, if you are happier thinking that this is a useless topic, no problem, be happy. Perhaps our Universe is the most useless thing ever created.

Juan-Esteban,

Well, if you want to deal with "meaningful" concepts, then each class is not only countable but is also of the generative kind, i.e. each object in the class is generated by the corresponding class generating system. And for such concepts there are no difficulties of the kinds that you anticipate .

Dear Juan-Esteban,

You misunderstood me, and I apolgoize if my last post was unpleasant, it was not my intention.

I do not think that your topic is not interesting, on the contrary.

But, as I tried to point out to you, your proof involves a philosophical position, even if you deny it.

The belief into a "Cantorian paradise" is involved by this sort of proofs, and Gödel believed also that Platonism is the correct philosophy of mathematics. So, you are in good company !

But it is true that I do not share this position, and that I believe that it is implied by the use and the abuse of classical logic in the field of classical set theory.

Dear Joseph,

You have said that "you do not share this position..."

I have no position. I simply have deduced a consequence from Cantor's theorem and neither I believe in this consequence nor I consider my will as an axiom. It is the deduction method what I am taking about. This is why I have claimed several times that science is a method instead of a doctrine. However, I suspect that philosophy is a doctrine collection without any method.

Dear Lev,

you have said: "Well, if you want to deal with "meaningful" concepts, ...."

Once again I must inform you that I "want" nothing. My will is not a mathematical axiom. Pure maths are not intended to describe real world. They are only a method.

This is why are called "pure"; otherwise are called applied instead.That is all.

Juan-Esteban,

I guess, I was misled by the form of your question to believe that you are interested in "meaningful" concepts rather than just set-theoretical ones. This is because mathematics, actually, doesn't address the concept of "concept".

Dear Juan-Esteban,

You wrote "I suspect that philosophy is a doctrine collection without any method."

If you pay seriously attention to the history of philosophy, you will be convinced that such a suspicion is unfounded. Every important philosophical work is a construction based on few principles and is like an axiomatic system.

It is not my philosophical reading of history of philosophy, it is an historical fact. Have a look on Jules Vuillemin's book: "What are philosophical systems?", it is a difficult but very important book that could maybe convince you that a philosophical work is systematic.

Now maybe you think about history of philosophy, as "doctrines collection without any method" ? I do not know if history is rational, I am not Hegelian. But no doubt that Hegel was a systematic philosopher, even if I do not share his philosophical options.

Cantor's theorem has the consequences that you have drawn in your proof only if one assumes the existence of the actual infinite, i.e. infinite sets (or/and classes). But there are mathematicians who do not make this ontological commitment, and it is not possible in my opinion to prove that they are wrong. Descartes himself - a good philosopher and an excellent mathematician - assumed that only God is actually infinite, the "set" of numbers was for him only the potential infinite. Again, this position belongs to the base of intuitionism, which is both a logico-mathematical shool and a philosophical option.

That is why I say that your proof must be relativized to a specific mathematical and philosophical commitment.

Best wishes

Jo.

Dear Joseph,

Consider two instances of axiomatic mathematical constructions, for example, group and topological space theories.

Group theory is based upon the following axioms.

1) There is an associative composition law.

2) There is a unit element.

3) Every member possesses an inverse.

From these axioms every theorem is derived blindly, simply, applying some game rules.

Now, consider the axioms of topological space theory

1) A topology T for a set E is a collection of subsets of E.

2) T contains with every collection their union.

3) T contains with every finite collection their intersection.

4) E belongs to T.

5) The emptyset belong to T.

Please, let me know the axioms of any philosophical theory.

Notice that the term axiom in maths does not denote any property of real world. It is only when you try to apply a theory to describe some part of real world when you must test whether or not real world fits into your axioms. This is why axioms are not considered either true or false. They are only pieces of a game and only the game rules matter, that is to say, the underlying method. In other words, axioms in any mathematical theory are not a doctrine, but a game rule set. I think that in any philosophical theory those claims that are termed axioms form a doctrine.

Take into account that the most abstract topological theories are absolutely of no use in order to describe any part of real world. The only aim is creativity, the art of abstract creativity.

Thus, I think that the term axiom means something different when used in a philosophical context or philosophy of science. The fact of stating new mathematical axioms is a risk-free action, because they are not considered either true or false. By contrast, there are always some risk in philosophical theories.

Best regards,

Juan-Esteban

Dear Juan-Esteban,

You asked me to let you know axioms of any philosophical theory.

Easy.

Pythagoreans: "Everything is a number".

Plato: "Everything is either an Idea or the image of an Idea".

Aristotle: "Everything is an individual thing (first substance), or a species (substance second)."

Spinoza: "Everything is an individual thing and abstractions have no existence in Nature".

And so on... You can check that theses axioms fit well with the respective philosophical theories of Plato, Aristotle and Spinoza.

Note that theses examples have the logical form of laws. Note also that they do not form together a consistent theory and that therefore you have to make a choice.

In mathematics also, we are not forced to believe that classical logic is the Core logic and we are not forced to accept the thesis that ZFC is the right set theory. As mathematician, you have to make a choice, so you take also a risk.

There are nevetheless indeed differences between mathematics and philosophical systems: the ontological system of the latter is of course larger and some philosophers do not accept the point that logic and science in general can be a mean to solve philosophical problems. No mathematician accept the claim of irrationality, some (bad) philosophers do.

Dear Joseph,

I appreciate the exposition of these axioms. I have learned something new. Thank you.

However, I must inform you that, as a mathematician, I need not chose any particular axiom until I do not try to apply it in order to interpret the real world structure. Nevertheless, trying to apply any theory is not a matter of choice, but experimentation. Every possibility must be tested and the last word is from real world, by no means is my decision, my choice or my will an axiom.

It is a widely used thinking method, to accept as truth those facts that one likes to be true. In this case, it is the human will the game rule, that is, the leading axiom. I think that this is the foundation of irrationality.

Best regards.

Dear Juan-Esteban,

You are welcome and the pleasure is mine.

I must inform you, as a philosopher, I need not chose any particular axiom until I do not try to apply it in order to interpret the real world structure. Nevertheless, trying to apply any theory is not a matter of choice, but experimentation. Every possibility must be tested and the last word is from real world, by no means is my decision, my choice or my will an axiom.

We agree ! ;)

Best wishes,

Jo.

Dear Juan-Esteban,

Actually, I cannot take issue with your thesis on the essential non-representability of some scientific concepts in a first order language. Nor can I take issue with what seems implicit in your argument: that the seeds of such non-representability may be seen to lie in Cantor's diagonal argument.

At a merely pedantic level though, what does appear to need clarification concerns your comment:

Of course you are right in observing that classical expressions of Cantor's diagonal argument do not feel the need to specify any method for defining the bijection between S and P(S), since the conclusion that no bijection between the two is possible does not apparently depend on such specification.

However, the importance of specifying a method can affect the specific conclusion that one draws from the argument.

By the principle of Occam's razor, the most that one may logically conclude from Cantor's diagonal argument is that there is no effective method (algorithm) of setting up a bijection between S and P(S). In other words, P(S) can be termed as algorithmically `uncomputable', in the sense of not being constructible finitarily.

In other words, we cannot conclude the existence of P(S) simply from Cantor's diagonal argument.

Concluding the existence of an `uncountable' P(S) within a first order set theory such as ZF on the basis Cantor's theorem---as distinct from Cantor's diagonal argument---is, of course, another matter entirely. The existence of P(S) follows straightforwardly from the existential axioms of the theory. Cantor's theorem only further identifies that P(S) has the attribute of `uncountability' within the theory.

Such a theory has no finitary interpretation and must, therefore, be seen as admitting Platonic concepts in any non-finitary interpretation; which, if accepted as definable, could perhaps provide an instance for your thesis that there are first-order concepts that cannot be defined explicitly within a first order theory.

However, Cantor's argument also has relevance to a first order number theory such as PA, which is not always extendable to a set theory such as ZF (see Theorem 1 in the link below).

http://alixcomsi.com/46_Non_Standard_Models_of_PA_Update.pdf

It is here that the assumption of the existence of `uncountable' sets may lead to inconsistency. Perhaps it was a similar need to distinguish between `uncomputability' and `uncountability' that led Turing to explicitly avoid appeal to Cantor's argument in his 1936 paper on computable numbers when outlining his Halting argument.

Bhup

Dear Bhupinder,

Your message is very smart. Thank you. But I have exposed my claim outside computability scope, in spite of one of my papers is related to computation. See,

http://www.i-csrs.org/Volumes/ijopcm/vol.5/IJOPCM(vol.5.2.6.J.12).pdf

Nevertheless, perhaps there are a lot of non-computable real numbers. For instance Chatin's constant.

http://en.wikipedia.org/wiki/Chaitin's_constant

Best regards.

Juan Esteban

Dear Joseph,

Axioms defined by Euclid in his Elements give rise to Euclidean Geometry which is very rich in theorems. For instance Apollonius' theorem: http://en.wikipedia.org/wiki/Apollonius%27_theorem

You have stated some axioms of philosophy, which I did not know. I suppose that these axioms have given rise to a lot of theorems universally accepted, like the Euclid's ones; otherwise these axioms would be useless proclaims. I think that the topic needs to be completed exposing some useful theorem derived from the referred axioms of philosophy. Knowing some theorems will help us to appraise philosophy better.

Best wishes.

J.E.

. " Henri Poincare said, "It is by logic that we prove, but by intuition that we discover."

Philosophy is the first phase of the rationalisation after an intuitive break through but the expression of this rationalisation has not reached the precision necessary for its axiomization. That later stage is called science when the empirical testing is done. The mathematics and science of today are the small part of philosophy which has mature to that later stage but the bulk of philosophy is dealing with extremely complex reality which still needs clarification. And when a mathematician instead of giving a proof decides that on the method of proviging proof in general, he is doing philosophy as all those that invented and are inventing the method of mathematics.

Dear Louis,

As usual, yours is a very smart and accurate answer. But axiomatic method is a machinery to prove theorems. The machinery to discover new topics is called heuristic. I think that a more complete chain would be:

Human-brain -> Information -> knowledge -> languages -> philosophy -> theories.

The last step can be carried out by illumination, by afectivity, by sectarism, by fashion, by logic.

If the device is logic, then axiomatic method is the only possible machinery. Take into account that any deductive chain either is infinite (impossible) or must start from some axiom system.

If if some branch of phylosophy is based upon axiomatic method, it must give rise to a theorem set. Otherwise, in philosophy "axiomatic method" means something different that I ignore. This is why I am asking this question.

Best reagards.

Dear Juan-Esteban and dear Louis,

That's a very interesting new topic, and I will be happy to participate. But just at the moment, I have not enough time. I have to finish a book in next days, and the deadline is very soon now. But I will be back.

Thanks for this dialogue.

All the best

Jo.

Dear Joseph

I want your book to be successful. I hope to hear from you soon.

Best wishes.

J.E.

Dear Juna-Esteban,

Thanks for this kind word. And thanks again for this interesting dialogue.

I will be back, no doubt.

All the best

Jo.

Perhaps the concept of ultimate infinity cannot be described by any language. There is already clearly an infinity of infinities (the infinite series of infinite aleph sets.)

Now it's simple to find an infinity of numbers that are not part of any aleph set, which leads to an infinity of infinities of infinities made up of aleph sets with such extraneous numbers added in (we already fail to quite precisely define the third different 'infinity' in that phrase.)

Working up from there we can define an infinity of infinities of infinities of infinities (rank 4) and with a bit of work we end up defining a rank ∞ .

I'm not certain we have encompassed all the extant numbers in this rank ∞ , but we certainly are nowhere close to having encompassed 'everything' (beyond mere numbers). It is not obvious that we can define what 'everything' is (to paraphrase Rumsfeld, there are things we know we know, things we know we don't know, and things we don't know we don't know), but trying to define the concept of all-encompassing infinity becomes quickly inexpressible by language, and unknowable: the very definition of the word 'ineffable'.

Chris,

11 : things we know we know

10: things we know we don't know

00: things we don't know we don't know

01: things we don't know we know

Dear Joseph

.....and Haraclitus of Ephesus "Panta Rhei".

Discovering things we don"t know we know seems to be at the basis of the socratic method:

quote from wikipedia:

"The Socratic method searches for general, commonly held truths that shape opinion, and scrutinizes them to determine their consistency with other beliefs. The basic form is a series of questions formulated as tests of logic and fact intended to help a person or group discover their beliefs about some topic, exploring the definitions or logoi (singular logos), seeking to characterize the general characteristics shared by various particular instances. The extent to which this method is employed to bring out definitions implicit in the interlocutors' beliefs, or to help them further their understanding, is called the method of maieutics."

Dear Isabella,

Yes, you are right. It is also a sort of philosophical axiom...

Dear Louis,

Tautologíes can only lead to tautologíes. In other words, they are the most useless claims.

Things we know we know is a tautology, there is no information in it.

If the term axiom means non proved claim, almost every thing in phylosophy is an axiom.

Dear Clara,

I cannot know what is your concept of definability. However if you are reading a text of mine, you must assign to the term "definability" the meaning I have explained in the first message of this thread. Otherwise you can understand nothing.

Word meanings depend on context and author. Mathematicians usually define each key word in our papers. I have used the concept of definability in a mathematical context and the proof is also constructed by means of axiomatic method and it is a consequence of Cantor's Theorem. If you have found some mistake in my proof I will appreciate your correction; but if you have misinterpreted my words....I hope you can appreciate my correction too.

Dear Juan-Esteban:

You wrote

************************** quotation ****************************************************

Since there are contributors refusing to accept the existence of concepts that cannot be defined, I think that a explicit proof must be posted.

Proof,

Let A denote the alphabet of any language L. Since A must be finite, there is a bijective map F from A into a finite set E = {1,2,3...n} of positive integers. Since every definition is a finite sequence of symbols in A, then there is an injective map F* from the set D of all definitions in L into N (natural number set). For instance, if p1, p2... is the sequence of al prime positive integers, for every defintion w1 w2 ...wm the assigned interger can be

F*(w1 w2...) = p1^(F(w1)).p2^(F(w2))....pm^(F(wm).

and by virtue of the Euclidean nature of the integer ring this map is injective. Thus, the map F* assigns a positive integer to each definable object. If the domain of F* contains the class of all possible objects, then there is an surjective map F* from the set of all possible definitions into the class of all objects. As a consequence the class of all objects is countable, which contradicts Cantor's theorem.

Summarizing, if Cantor's theorem is true, there is non-countable set of objects that cannot be defined by means of any language generated by a finite alphabet.

********************* end of quotation **********************************************

When you further write

'If you have found some mistake in my proof I will appreciate your correction'

do you refer to just the text reproduced above, or is there a text that I have overseen and that is part of your argument ?

Please make explicit why the set of all concepts defineable in L can not be countable. Is your argument that any finite set of concepts has again be considered a concept? (I seem to remember to have read this argument some where in this huge discussion but can't find it anymore.)

Dear Mutze,

First, I must point out that science is a method not a doctrine. Is the proving method what you can either reject or accept.

I do not care whether or not there are concepts that cannot be defined. What I have shown is that if Cantor's Theorem is right, then there are concepts that cannot be defined by means of any language.

You ask me why there are non-countable classes of concepts. Do you seriously are asking that? Cantor's theorem is essentially the proof that the collection of all subsets of any infinite set cannot be countable. Likewise, the real number set is not countable.

Your question must be asked to G. Cantor. By no means I feel myself responsible of his theorem. However, if Cantor's theorem is right, it is a straightforward consequence of it, that there are concepts that cannot be defined. It is so simple the proof that I cannot understand how there is any understanding difficulty such a elementary deduction.

Best "uncountable" regards.

Juan Esteban-

Dear Juan-Esteban,

I'm back :) .

You wrote :" if Cantor's theorem is right, it is a straightforward consequence of it, that there are concepts that cannot be defined. It is so simple the proof that I cannot understand how there is any understanding difficulty such a elementary deduction."

It seems to me that your proof is flawed just because you make an illegitimate correspondence between the extentionalist point of view and intentions: "concept" has a meaning, and of course there is also an extension of "concept", and an infinitary one. But it does not mean that you need an infinite alphabet to understand what infinity means. "Definition of concepts" is exactly in the same situation: the existence of uncountable sets does not mean that there are concepts that cannot be defined. We know that there are an infinity of "unspecifiable real numbers", but they all correspond to the same concept of real numbers. I can give still a more simple example: we know that there is in our universe an infinity of "things" that we will never know, but it does not mean that they are things which do not correspond to the concept of "thing".

I believe that this point explain the difficulties of some of us in this thread to understand your proof: you are mixing intensions and extensions, and, pace Quine, it is a dubious philosophical strategy.

All the best,

Jo.

Dear Joseph,

On the one hand, you have biased my claim. I am talking about definability "by means of a language," and you are dealing with some kind of abstract definability.

Classes containing non-definable objets can be themselves definable. For instance the class of real numbers can be defined easily, however it contains some members that cannot be computed (Chaitin constant) and others that cannot be defined.

Likewise you can denote the whole universe; however, surely, you cannot denote all objects lying in the Universe.

On the other hand, as I have mentioned early, the meaning of each word depends on the context and author. If you assign meanings to the key words different from what I have defined, then you are arguing against your own interpretation and your speech is a soliloquy.

I have defined the term language algebraically as a partial free-monoid generated by a finite alphabet. Then I have shown, through a Göedel-like numbering function, that every language satisfying this definition is countable. To assign a definition to each object in a class C, from an algebraic viewpoint, consists of defining an injective map from the language into C. According to Cantor's theorem such a map cannot be surjective. That is all. If you analyze my proof you must assign to each term the meaning that it has in algebra and I have defined. In addition, you must work by a mathematical method,

Nevertheless, if you swim better inside philosophy, we can try to define only one term, for instance "truth" or "God" and you will see that the thread becomes an endless one, giving rise to an infinite sequence of different definitions running in a Plato-like dialog. Perhaps the only conclusion would be that everyone assigns a own meaning to each of these words; therefore they are not possible to be defined in spite of Cantor's theorem could be wrong.

Plato's dialogues are based upon the impossibility of defining accurately concepts outside maths. This is why some contributor tried to provoke such a dialog with me in order to enjoy leading to no aim. But it is only possible in colloquial or natural languages. Fortunately, algebra has a different structure. Algebra lovers are always protected.

Best regards.

J. E.

Dear Juan

You just wrote: "That is all. If you analyze my proof you must assign to each term the meaning that it has in algebra and I have defined. In addition, you must work by a mathematical method,"

In your proof, the conclusion "there is non-countable set of objects that cannot be defined by means of any language generated by a finite alphabet." is a definition, i.e. the definition of "non-countable set of objects that cannot be defined by means of any language generated by a finite alphabet", which is itself done by a finite alphabet." But I do not know the meaning that such a definition has in the algebra that you have defined.

Does it make sense to ask you to tell it? I do not know. Could you help me please?

Best regards ,

Jo.

Dear Joseph,

I apologize, my claim needs some explanation.

It is a well-known result of Cantor's set theory that the finite union of countable sets cannot be uncountable. Thus, if E is an uncountable set and the subset D of E consisting of all definable members in a language L must be countable, then its complement U must be uncountable. Indeed, the complement U consists of all members that cannot be defined in L, therefore every uncountable set contains an uncountable subset of objects that cannot be defined. I have considered that this fact is obvious, because it is a straightforward consequence of Cantor's theorem, but it is only obvious by who knows this theory.

Once again I must inform that I do not care whether or not every object is definable by means of some language. My only claim is that, if Cantor's theorem is right, then every uncountable set must contain an uncountable subset no member of which is definable.

Best regards,

J. E.

Thank you Joseph for your kind words.

Follow your arguments and.........enjoy it immensely.

The concepts of...expressivity by pure formulas...got my

full attention...........but what about...the importance of poetry

in philosophy...would it help to pass the "defnability" deadlock?

Nietzsche saw philosophy and poetry as mutually beneficial

pursuits...and contemporary thinker...Raymond Barfield

in his "The Ancient Quarrel between Philosophy and Poetry"

insists (up)on the idea that philosophy's concepts...have been

heavily influence by poetry.

I would never eliminate or/and diminished the importance of poetry

in my life's philosophy..................what about you?

With warm regards Isabella.

Dear Juan-Esteban,

I thank you for this very clear explanation. Misundersandings came from the gap between the question for this thread and this conclusion of your proof.

Best wishes !

Jo.

Dear Isabella,

Thanks for your kind message also. I like also some Nietzsche's books. And poetry, especially Appolinaire and Verlaine. But because it is not the topic of this thread, I do not develop here.

Warm regards too,

Jo.

Dear Juan,

I asked you for your reason to conclude that there are uncountably many concepts. I know that you appeal to 'Cantor's theorem' here, but I would be interested in knowing exactly how you apply Cantor's theorem to concepts. Which set together with its power set is considered? (Please understand that it is not Cantor's theorem itself that is under discussion here.)

For me it looks reasonable to assume that a concept can be communicated by a finite string of characters. How else concepts could play an active role in our thinking? For concepts understood in this way your proof would obviously not hold.

I think Clara was right when asking for a clarification of your notion of 'definition'. In addition, I ask for a clarification of your notion of a 'concept'.

The two questions seem to be closely related: If you can give convincing reasons that a 'concept' needs not be representable by a finite string of characters, than I can probably turn this into an argument that also not all definitions need to have such a representation.

Dear Mutze,

I cannot guess which is the meaning you assign to the term concept. See the definition in Oxford dictionary.

----- Oxford Dictionary -----------------

Philosophy: an idea or mental image which corresponds to some distinct entity or class of entities.

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

If a number is an entity then there is a concept or mental image, which corresponds to each number. If there are uncountable real numbers, then there are uncountable corresponding concepts (mental images).

Nevertheless, I suppose that you are asking how a concept that cannot be defined can exist.

If so, you are misunderstanding my messages. By no means I have said that there are concepts that cannot be defined. What I have shown is that there are concepts that cannot be defined "by means of a given language."

The algebraic proof is simple. A language is a partial free monoid. A partial free monoid is generated by some subset, each member of which consists of a single word and cannot be defined, its meaning is assigned by convention. Once a meaning is assigned to each member of the generating system, the remaining definitions consist of word sequences, therefore only a countable set of definition can be built.

I suppose that you will interpret my words as if they were in a philosophic context, but I am working in the kingdom of algebra. Thus, arguing outside the kingdom of algebra is not an adequate method.

Some illustration can help you to understand my claim.

Consider a sub-language L of English consisting of three words, for instance John, Peter and Jeremy. A family with four sons assigns a name to each child using the sub-language L. Indeed, it is not possible. There is always an unnamed child, because only 3 names are available, while there are 4 sons. You cannot know who will be the unnamed child until the family takes a decision. Nevertheless, before any decision, you can know that there will be one unnamed child. However, without knowing which is the family decision, you cannot either describe or define the unnamed child.

The structure of my claim is similar. When you assign a meaning to each phrase of a language L, there is an uncountable set of objects that cannot be defined in L by a word sequence. Suppose that O is an undefinable object in L. You can change the assignation obtaining another language L0 in which O is definable, however there is also an uncountable set of objects that cannot be defined in L0, and so on. That is to say definability is not an intrinsic property of objects or their concepts. It is a limitation of finitely generated languages. For every language L and each undefinable object O, you can always obtain an extension L0 of L in which O is definable; however a language containing all possible extensions is not allowed by Russell paradox. I have shown this topic previously and I need not repeat it.

Finally, an infinite word sequence can be handled when can be finitely defined. For example, the rational number 3.45454545… can be finitely determined, for instance, saying that the two figures 45 are endlessly repeated. However you can build a real number 4.10 00 001…by tossing a coin. The nth figure is 0 or 1 depending on the result of the nth toss. You know that this infinite process gives rise to a real number, but you cannot either define it or determine its infinite sequence of figures by any finite process.

Best regards.

Dear Juan-Esteban,

You wrote, in reply to Ulrich:

"If a number is an entity then there is a concept or mental image, which corresponds to each number. If there are uncountable real numbers, then there are uncountable corresponding concepts (mental images)."

But the premise is false!

Numbers are certainly not entities, but classes of equivalences, or functions, not entities at all... In other words, it is an abuse of language to speak in the language of classical set theory in order to say:

$ 1 \in N $ in order to "prove" that number 1 is an entity . That's not a "proof", but just a way of speaking.

Neither numbers, nor concepts are entities. I am , with Ulrich probably, physicalist on this topic.

Best wishes,

Jo.

Dear Juan,

thank you for your extended explanation.

If you accept each real number as a concept then, of course, you have uncountably many concepts and only countable many definitions in a single language. For a trained mathematician this is a single thought and doesn't need further explanation.

Unfortunately my brain is too small to house images of 'actually infinite' objects, if they are not build according to constructions that can be described with finitely many words. So your 3.4545... , and pi, and Chaitin's constant are ok, but nothing that would need more synapses than I have. So the simple situation is:

1. Your notion of 'concept' lacks any plausibility for me.

2. After having given a definition of 'concept' (as you now did) you are free to formulate theorems about concepts and proof them (as you successfully did).

3. Whether such a theorem is interesting or banal depends also on the definitions of its ingredients.

Dear Joseph,

Once again I appreciate your smart message. Once again I must say that you are biasing the question towards a semantic one. You say that number is not an entity but an equivalence class. I have to repeat again and again that words denote equivalence classes. But equivalence classes have instances, which are entities.

For instance, the number 2 is an equivalence class; however, these vertical lines | | form an instance of 2 that has a mental representation in my mind.

My proof deals with cardinalities, and dealing with cardinalities you can make an abstraction of every other attribute. In addition, in spite of being equivalence classes, numbers can be handled by means of their instances that are entities. Even the irrational number √2 can be represented by the hypothenuse of a triangle.

This is why numbers can be handled as entities provided that only cardinality matters.

In any case, equivalence classes can be also regarded as entities. Nevertheless, it does not matter. If you negate that equivalence classes are entities; therefore they cannot have a mental representation, then there is no entity in the Universe.

You yourself are an equivalence class. While you are reading this message you have lost some atoms of your body. Your hair has enlarged a little. You are not the same thing from a second to another. What is denoted as Joseph Vidal-Rosset is an equivalence class determined by those properties or attributes remaining unaltered under the continuous transforms that your mind and your body are continuously experimenting.

Since what we denote as Joseph Vidal-Rosset is an equivalence class, according to your claim, you are not an entity and the concept of Joseph Vidal-Rosset cannot exist.

I can imagine that every word either is a function or denotes an equivalence class. Even there are programming languages, like Lisp or Haskell, that only handle functions. Every expression in Haskell is a function and it is a language for general purpose.

Finally, let us consider the word “God.” For a Muslim denotes a unique person. For a Christian it denotes three persons. For a Pantheist denotes the whole Universe, and for an atheist denotes a non-existing being. Do you agree that the term God denotes an equivalence class? But a very large equivalence class, because the meaning that an atheist assigns is far from the one of a Muslim.

The core of your claim is semantic, and word meanings are context and author dependent. If you put any of my sentences in other context you can find always errors. This is why with a good behavior; my algebraic claims must be analyzed by means of an algebraic method. Nevertheless, when a reader asks me for some explanation, I have to write in a colloquial language, which is always ambiguous. You cannot take an illustration as a proof. My proof is algebraic and in algebra a language is a map from sequences of words into meanings. Every argumentation outside the map and set theory does not fit into my speech.

From now on, I will not answer any question in a colloquial language to avoid this kind or misinterpretations. If some readers do not understand algebraic methods, it is not my problem.

Best regards.

Dear Juan-Esteban,

I appreciate also your smart defense of your proof !

You wrote: "Once again I must say that you are biasing the question towards a semantic one. " Really? I claim that if your proof is valid in FOL, therefore, via FOL's completenss, it is both syntactically valiid AND semantically valid. Therefore semantic approach is legitimate.

You wrote: "For instance, the number 2 is an equivalence class; however, these vertical lines | | form an instance of 2 that has a mental representation in my mind. "

But I deny that a mental representation is an entity.

You wrote: "If you negate that equivalence classes are entities; therefore they cannot have a mental representation, then there is no entity in the Universe."

Suppose that human species one day disappears and that thought is a phenomenon depending only on our brain activity, I hold that if human disappear, disappear also "equivalence classes", because this expression denotes only thoughts which are useful in science. But I deny that there will be no entities in the Universe.

You wrote: "You yourself are an equivalence class. " I deny that I am an equivalence class. I am an individual object, an element, not a singleton.

I am afraid that you make a confusion between grammar (or logic, or mathematics) and reality. You should read Kant in order to avoid this dogmatic and metaphysical confusion. I am not an equivalence class, because I have a spatio-temporal existence which is easy to define: a first day, more than 51 years ago and one day in the future a last day. Concepts and meanings also have an existence, but they are not substances and how they exist is more difficult to describe.

Best wishes,

Jo.