1,2,3, ... - Ethics!
a 'stupid' question: what are the natural numbers? So the count number 1, 2, 3, ... To learn each infant is able - that is to be a philosophical question of weight? And have nothing to do with ethics?
Answer: Yes.
It is, inter alia, the importance of such concepts as "infinity", but also to the understanding of "truth" and some others in our culture and science, but also in everyday life and the ethics most important words, yes even about the term "freedom". It is only apparently, only to glimpse, just a matter of mathematics. In fact, we are dealing here with the basic problem of all science, a problem that in the first half of the 20th to solve the century of the philosophers and philosophers of science has been tried - but it now seems to have wiped resigned to the side.
What is the problem?
we describe an example from mathematics, it is the "continuum hypothesis" mathematically expressed little amateurish, they will argue that the set of natural numbers (ie the numbers that we are: 0.1, 2.3 ,.... ) No less and no more "powerful", no less, but not more extensive than the set of real numbers (which are the decimal numbers, eg 1.1234, 3.1415926; 50000.48723094823742743 ...)
So if you both numbers' by one, each element in each case with a number of "cardinal number" names, then the continuum hypothesis asserts that the two are not different Incremental extensive. Layman's considered one thinks easily that of course it must be more real numbers than natural numbers because they add a comma every natural number and then you can write any number of x-integers and sequences of numbers behind it. Only by adding a comma to a natural number can be from their endless - make real numbers - 'without end'.
This problem has actually been known from antiquity as the paradox of the race between Achilles and the tortoise: The tortoise gets a head start, and after the start signal Achill collected only once on the half of the projection, while the turtle has also been a distance traveled, garnering the Achill half still needs time during which the turtle back a bit ... etc. Known to resolve this paradox is easily once you have invented the calculus. Now you need but the number of "infinite" - and the question is: is infinity a number? Is it the "actual infinity"? Or should we speak of a "potentially infinite", then proceed from the idea that you can always count on, without end? Again, this dispute has lasted since ancient times. Aristotle chose the potential infinite, the Platonist, for example, St. Augustine, saw the Update-infinity as divine.
Kurt Gödel and Paul Cohen showed that the continuum hypothesis within the set theory is neither provable nor refutable, and so take them because you need it, as unproven "axiom" in the set theory on (Zermelo-Fraenkel with the continuum hypothesis, "ZF + CH"). "Unproven" means: you can not say, the axiom is true, yet it is wrong. If you want to "prove" an axiom, or legitimize its use, one must resort to a 'more than gerd marked' theory. And we will review these "meta-theory, we are facing the same problem of the legitimacy of their conditions: it needs new meta-theory of the next stage and so on and so forth. Often the fees were waived to enter such a never-ending staircase and appeals to "reasonableness" of the assumed Assumptions, the axioms. It is overlooked, however, that even the explanation of why you hold this or that assumption is plausible, indeed even the wording of the adoption alone, on theoretical relationships based - even unreflective everyday experience and common places are provided, they hide in the ' natural 'vernacular. It is therefore based on pre-judgments, sometimes to the anticipation of a desired or 'useful' deemed income of the relevant theory.
you can possibly escape the problem by proving that the amount of these theories stepped in itself is complete and consistent? - That would be the prerequisite that it would assess each case to be obtained conclusions from this theoretical step pyramid as "true" or "wrong", and that as soon as they are in life, in art and industry, as well as in economic policy or the law applies to produce reliable, what you have "proven" argument. In "real life" if one is not exposed to the method of trial and error. " At least when it comes to the lives of others is involved, that would be seen immoral - this anticipation on the methodological grounds of ethical statements would I allow myself here.
By Kurt Gödel's famous incompleteness theorem " (1931) was aware of this problem in the establishment of theories according to the meta-theory concept as intractable: the quantity and type doctrine that had developed Bertrand Russell in support of mathematics can not keep their promise. Mathematics has a 'black box' at the very beginning of their methodology. You do not know what's inside, as in the Losbude takes you into it blindly and works with what you have caught. How can we then ever be sure if the compiled results of such theories are true, correct and reliable? The mathematician Leopold Kronecker had it in the 19th Century, complained: "All the figures of the Lord has made, everything else is the work of man." Mathematics students should spend their first semester as in the theological faculty? Or should we rather not have to worry about the scientific justification of the integers?
have in exactly this way all sciences, not only mathematics this black-box problem as long as you have not solved the problem of justification for their respective sectors. For example, in psychology, "intelligence is what the intelligence test measures" - that is really an intelligent initial definition? Or in the economy: What is "money"? What is "capital"? What is "market"?
etc. It is the genuine task of philosophy, such mere plausibility on the terminological basis to replace the words at the beginning of the various sciences by objective method of scientific reasoning .
The problem is justification from the usual approach: to design a theory by defining first term, that is claimed by ordinary language statements that defines technical terms and rules to be developed with those from the initial claims to knowledge. Then you invent a second theory on a "higher" level, with which it checks if the first theory is closed and is free of contradictions. Only if these two conditions are met, we worked on the conclusions . Trust Now raises the question: What about the Vertrauenswürdigket the theory of second order? So you invent a theory of third order, with which one examines the theory of second order for completeness and consistency ... and so on, ad infinitum. Assuming the theory, say, 10 Procedure states that the theory 9th Order is invalid - so the whole house of cards collapses. In the Gegenwwart we meet everywhere the proposal to move to a different level kausalistisch. If the Finanzsysstem in the crisis, are now to the behavioral sciences in support of the economy are taken into account and deal with the behavior of stock junkies; for behavioral research is currently popular again, the reference to genetically anchored thinking and behavior. How the one about the origin and the numbers in any "number of genes. But remember: the genetics would be to establish methodically ... and then there is also a sociology of science that deals with the question of why researchers deal with this or that topic - what about the Wissenschaftssoziolgen ...? etc., etc.
The philosophical justification for the solution strategy problem is: you have to check with each own argument means of a theory, even if it is complete and consistent. The theory will can be justified on its own. There lies the crux: Godel's incompleteness theorem says you can either one or the other - but never both together prove both the integrity and the consistency of the theory with their own resources.
is the consequence: it is referred to the belief that the unfounded - arbitrarily given initial claims of a theory "somehow reasonable" are. And only on the basis of this belief can be "trust" the conclusions. Scientists come to its status of the priest, because the justification of science itself does not follow scientific certainties.
This is no small problem - a big problem is that the religious foundation of the concept of truth is ignored. An example of the largest machine in the world, the Large Hadron Collider near Geneva. In the experiments to be created black holes that you, me, the whole of humanity, the world and also the LHC into nothing that could suck in a singularity. Some who know more about physics than me and probably you too, express the fear that the LHC could be a Doomsday Machine. Others who understand of physics is also more than you and I and the greater part of humanity claim however, this fear is unfounded, possibly ridiculous. For none of these allegations, there is a notarized proof of the truth, it would probably be difficult to find a qualified notary. After all, is the probability that the "estimated" the emergence and continued existence of small black holes by the experimenters, (1: 10,000,000), is many times higher than the probability of a jackpot in the lottery (1: 140 million).
is alarming, not even the highest German court was empowered to rule on this right to the divine lottery, leaving the fate of the world so completely, by a state immunity CERN outside any jurisdiction (most amazing! see video!) active "free" genius of the scientist.
Once before, in the testing of the first hydrogen bomb, there were fears it could be the atmosphere on fire. Let us hope that this time too well.
Is it the existence of the world's thirst for knowledge, the plausibility of assertions and trust of those who are enthusiastic about their experiments? Even if you doubt their sincerity and seriousness in any way would like - it does not lie in the egomaniacs of the Geneva researchers, but at the axiomatic basis of their science, thus ultimately to mere plausibility. Should we as scientists - believe and hope - and as a neighbor of scientists?
What now - is the science, understood as an instance that the creative life of human-compatible with each other peacefully helps design that is under the principle of universality, to save? Like the freedom of science is to be understood? Do I have it, again hierarchically put under guardianship of security guards - Or is something wrong in the "solipsistic" and "individualistic" concept of freedom? What has the freedom of the individual with the need for public enlightenment?
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The philosopher and mathematician Paul Lorenzen has developed in its "meta-mathematics" (1962) is an important solution for addressing the problem of justification of science, on the grounds of their ability to generality. If this solution is focused on mathematics, they are still the guidelines for the solution in other sciences. The basic idea Here: a terminology of 'down' supported, ultimately from a pre-linguistic region, out of practice. One must then not to a preceding, who left their overall level of language theory.
The intention Lorenzen on his constructive approach of the "meta-mathematics" was to show a way out of the dilemma that was created for the sciences by Gödel's theorem. This solution should not be axiomatic - for the axiomatic approach is precisely the reason for the epistemological dilemma of the incompleteness theorem. "Axiomatic" you can undefine the problems with the continuum hypothesis easy - you just have so only through the Axiom of choice to set theory to accept that the world is for the mathematicians seem right again. The heist yet: after not within the set theory (Zermelo-Fraenkel) the continuum of real numbers can be proved, it is added, by axiom, as indeed plausible, but as unfounded and unprovable condition - so to speak, point, finished! 'par ordre du mufti'. This is the philosophy entirely unsatisfactory to the axiomatic approach, the generality of theories constructed in this way based solely on the kindness of their conditions.
The methodological principles of a "constructive" Approach à la are Lorenzen: go step by step, without compass-like reference to words, that, in ordinary language, already are 'on the tongue', but have not yet been specified in the order of the building down in its use. In the case of mathematics, these first concepts - the count number. After putting the formal logic to the further construction of arithmetic can be performed easily. (Incidentally, the logic required for this - more precisely, predicate logic first order - you can count on, because it meets the criteria of Gödel's completeness and consistency of their own criteria)
So what are the Count number 1,2,3 ,...?
Since mathematics, like all sciences, not with the everyday language, but with a precise technical language ("Ortho language") works, it must realize that their artifacts, knowing what they're doing produce, synthesize. Of course, one uses it in everyday language, but only for explanation and description of what you're doing on the blackboard in the lecture or laboratory - the start tag and its content is important to the science, however, must be defined in a way that " principle "requires no substantive use of the meaning of everyday words. So, as would almost speechless People 'ab ovo', fresh from the egg hatched the stork, develop their own artistic language with each other. That sounds difficult, so the philosophy seems to have the difficult task will be drawn as Munchausen's own bootstraps.
But it is also very simple. Lorenzen suggested the following guide to action in order to construct the count number:
Mach I a line, then another to II to III, then another, and so on. Generally speaking, follow the instructions:
=> I, n => nI
However, there are philosophical critique of this "abstraction" from the objectivity of Zählzeichen: it is not clear enough in relation to the philosophical and educational requirement for "universality". This is also related that the design follows the number one rule - whoever to these, and can they be other reasons, except that it seems trivial, of course? What is different, philosophically, this generally positive from the Peano axioms that too - criticized by the Constructivists in the "axiomatic" approach - Rules for the definition of "successor" are? The axiomatic justification for Lorenzen is therefore not in question because the formulation of the axioms is arbitrary, unfounded and relies on the natural language as a metalanguage.
To answer this question, I would rather speak of "abstraction" of "generalization." Abstract say so, leave out something. What? Who makes such decisions as necessary to do it based on the relevance of being cut off and the transmission of Bela?
In generalization, however I understand the change in perspective away from the object toward the subject, the repeal of the subject-object dichotomy. The object is not abstract, but the subject is generalized and its relationship to the object - make it a win, all subjects of common, object. The activity in this case the activity of constructing the counting numbers is to be carried out by people who make their relationship to their discourse partners themselves so that it makes no difference which carried out by the participants which discourse step. That is, they should be represented, represent each other can. You could, without affecting the result of one of the parties changed, such as draw lots who should take what step. Only when we have completed that step of generalization of our views, we can rely on that we produce words and distinctions "objectively" are - that is, it is equivalent to whether you or I or someone else receives the invoice or the reasoning carried out - it is understandable by everyone (and checked for logical errors, to correct you then agree could).
To do this we do the work of constructing figures by lines in an interaction perform the roles of those involved, according to certain rules, distributed and exchanged. The starting point is this: one, on the initiative side, exerts the construction activity, the other, on the Responsseite, monitors compliance with the design rule. There is just one construction rule, this is the discourse participants, each of which is on the front page of initiative, what he has to do. It says:
"Do the same thing that your predecessor has done," . It remains valid for all participants.
You see, this is not "material" required. It is a general rule that applies equally to everyone, and compliance is demanded by their counterparts. "Materially" only provided that it is making around the lines, which could also alter, for example: apples in the basket - it is about the merging of any discrete items.
We will be surprised to see that, depending on the rules give the position distribution in the discourse completely different number systems - although only follow all the same design rule. We will then get different number concepts.
This means the structure of the social order, and not the design rule determines the number concept!
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Let's start with the first game of discourse, in which each step is performed by another participant.
you play through the discourse in a transitive hierarchical order: the first in the discourse with the second, the second with the third party, etc. The cycle is closed when the first party once again takes the initiative and plays with the last player. The first initiator thus takes a prominent position: it determines the beginning and end, and he is twice as active on the initiative side. In contrast to all other participants is he was never on the response side, are against all those once on the response side and once on the initiative side. These are the rules of the transitive relation. The first initiator
determined that sovereign the beginning and the end of the resulting sequence of numbers, their length. Whether they are "infinite" will depend on his caprice.
The work instructions to the respective owners of the initiative side is, as others have said:
"Do the same thing, what did your predecessor."
has the same order for each participants 'rights', none is an advantage or disadvantage when the game began. All are equal, and so can not complain. This is a fundamental ethical principle that the results can also be regarded as binding for all participants.
The initial step looks like this: The first
initiator finds a situation that is empty. This he adds a dash. Result: 1
Since the first initiator has no predecessor, he is free to choose what he does. He could have made even more strokes, he could lie down gravel, which is also always think of him - he is in this position, the first initiator, sovereign. But we want to be satisfied with the minimum output.
Thus the first initiator of the discourse, however, characterizes the activities of its subsequent players. Since there is no rule which player should take the first position, this 'advantage' at the first player irrelevant. He is not justified - because before the start of the game to generate the numbers, the participants in these 'number game maker' has not even count - and therefore also not be attacked or refuted - he acts like an axiom. Another result is thus already apparent: this "axiom" determined - due to the logic of transitivity - that the same, which determines the beginning, determined the end. In it lies the determination of the "infinity". We want to alphabetical
the first player therefore proposes the Number of participants in the discourse and make him call DAM.
The next discourse participants ADAM, who is moving in the first step discourse on the (controlling) response side was now on the initiator side.
He does the same thing that his predecessor, DAM has done in its passage: it adds the zero (the initial number of the predecessor) and one (the result of the previous number) together - Score: 1 Generally speaking, it adds the two numbers (source and result number) of its predecessor, the result is his newly formed number This new income figure, along with its original number is then joined by his discourse successor again, bringing the new number of discourse successor formed.
comes in the third round now Bedam who has monitored the second round on the response page on the initiator side, and he does the same, what did his predecessor: it adds the dashes (output and result) of his predecessor (here: ADAM) together. Score: 2
In the fourth passage comes CEDAM, the Bedam monitored on the initiator side, he does the same, so did his predecessor CEDAM: it adds the lines of his predecessor, here: BEDAM together. Score: 3 So the game goes on - without any discernible, bennennbares end. It can 'infinite' number of discourse started with 'infinite' number of participants will be carried out - until the first initiator Episode flush closes by taking the initiative to date to the last participant.
compiled in an overview:
a) The discursive production of the hierarchical numbers (Fibonacci series)
The is design provision for the initiative page
"Do the same thing that has made your predecessor"
The Respondent monitors critical of the implementation of this provision, characterized by the question mark "?".
; Transitivity
activity Respondent initiator result
ADAM DAM number
default 0
?
default + addition 0 + I => I 1
BEDAM ADAM
Vorgabe I
; ?
default + addition 0 + I => I , 1
(as predecessor)
; CEDAM BEDAM
default I
?
+ Added default I + I => II 2
; (As predecessor)
Dedam ; CEDAM
default ; II
?
default + addition ; II + I => III 3
(as predecessor)
EFDAM Edam
Vorgabe ; II
?
addition default + II + III => IIIII 5
(as predecessor)
. .
........
........
.......
........
........
.......
Ω-DAM DAM
default + II .. IIIII ...
; ?
default + addition ; II .. IIIII + ... =>
(as predecessor) IIIIIII .... Ω
It is clear the "Fibonacci Sequence" with the numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ... ...
numbers as concepts
We now want to change the discourse play in relation to the exchange of positions usually the discourse participants. The object of the game, the action instruction to the respective players on the initiator side, however, remains as in the previous game:
"Do the same thing that your predecessor has done!".
The exchange of positions now generally requires that each exchange two discourse partner in two discourses aufeinader following their positions in a symmetrical manner. Everyone is so even with the same partner in the initiative, and once in the response position.
The generated social symmetry between the discourse participants solve the identity of the Order of discourse participants with the order of discourse steps. Now it may occur in the transitive Dirkursdurchgang all discourse participants in any discourse in any step, because the players themselves, as a result of the generated social symmetry between them, could represent each other.
If the classification in the debate over how precisely determined solely by the rule of transitivity, then the result of the activity of each Teinehmers, depending on the ranking point where by he led his "turn", is different from that of any other - even though he has followed the same rule as all the others: "Do the same thing that has made your predecessor!" The seems surprising, each participant is given by the Fibonacci number generated by it identifiable. The order of occurrence of a participant in discourse is not predetermined - it can be determined by the self. In this regard, there is no rule, the alphabetical order of our players Adam, Bedam ... should not suggest such a series. She is only half the presentation, as part of the explanatory "Para language" is selected, the I, the writer, you, the reader performing, and paraphrasing what is being played.
This creates an amazing difference in the resulting sequences, and this difference depends only on whether the "discourse ballet" the social Symmetry of the participants generated or not and the transitive order of discourse is carried out under the condition that they may represent each other (which is not a requirement). The discourse preceding symmetry we may call " dialogue."
Assuming the symmetry of the transitive social hierarchy is divided among the participants. The first initiator is therefore a no privileged position, either at the beginning, at the end. On this aspect, we will later come back as very interesting.
b) The dialogical generation of numbers: natural numbers
The dialogic design provision for natural numbers is as above also
"Do the same thing that your predecessor has done!"
The Respondent monitors critical of the implementation of this provision, characterized by the question mark "?".
; SYMMETRIE
Respondent Initiator
BEDAM ADAM
; ; ; -
?
I
ADAM BEDAM
; ; -
?
I
follows from that of social symmetry and ADAM BEDAM to their mutual representation ability. These first two symmetrical discourses each of them made a dash. Now if one of these two, for yourself or in representation of the other, in the following transitive passage into the next debate with CEDAM happens, he will, no matter acts whether as ADAM or BEDAM, draw a line - in compliance with the rule "Do the same as a parent". For now in force because of their social equality is the action of the first actor the decisive action, the action reference for all successors, that this self-generalization have undergone.
now arises, but also a symmetrical relationship with the CEDAM. For instead of the transitive discourse sequence (without prior symmetry between ADAM and BEDAM) with the unilateral position distributions in which ADAM exclusively before dividing on the initiative side, and CEDAM only the passive Responsseite occurs (note the shaded positions!):
; position distribution
; transitivity
(without prior symmetry)
Respondent initiator
BEDAM ADAM
; CEDAM BEDAM
CEDAM ADAM
arises now because of the mutual representation of Adam and BEDAM in the first two rounds of the transitive discourse following sequence change in relation to the position taking CEDAM and Adam:
Position distribution
transitivity
(with prior symmetry)
Respondent initiator
CEDAM A-B-DAM
A-B-DAM CEDAM
... Etc. .. ... Etc. ..
this means now is the symmetry between ADAM, BEDAM and CEDAM in the context of transitive sequence "automatically" produced, can from now on, these three represent each other - so that sets this Verallgmeinerung the discourse participants continued to any other envolved Player:
Respondent initiator
CEDAM A-B-DAM
A-B-DAM CEDAM
DEDAM A-B-C-DAM
A-B-C-DAM DEDAM
... Etc. .. ... Etc. ..
CEDAM will be just as capable of representation and of representation over its predecessors and ADAM BEDAM, so that CEDAM then adds in his discourse on the initiator side following a stroke. This continues then at Dedam etc.
The symmetry between the discourse participants is now so irrelevant, how many strokes were each as a starting point in a discourse passage at the beginning - the rule: "Do the same as your predecessor," has' materialized "to" do add a dash ". So now we have a justification for the successor design provision Lorenzen n => nI.
Because of the resulting general mutual representation ability and power of representation, we can now, in a wider discourse transitive course, the names of the players leave, and limit ourselves to indicating the position of "initiator" or "Respondent". The players play as a "general person" N, so that one could actually omit this designation.
production of natural numbers by
; Transitivity
implemented in symmetry
Respondent Initiator
NOMEN NOMEN Score
n
?
I nI
NOMEN NOMEN
of
?
; ; I nII
NOMEN NOMEN
nII
?
I nIII
...... etc. ...
Through the production of social symmetry between the discourse participants through the course even created a discourse that produces the sequence of natural numbers.
So this is evident a significant difference from the only transitive discourse choreography that results in the sequence of Fibonacci numbers - even though the instruction to the participants in the discourse! "Do the same as your predecessor has made" the same in both cases and no difference defined in advance.
The difference is particularly visible in the "infinite": The individualistic generation of the sequence was the primary initiator of the end of the DAM Episode jurisdiction - whenever it wants to be sovereign will. The other participants ADAM, BEDAM ... can not stop the episode. Hence the idea arises of what it could be the end, as something that exists in the will of the DAM. The direct analogy to the Creator God is evident. So we can also understand why Aristotle argued for the potential infinity: he had replaced the personal God intended by the "first mover unmoved" (πρῶτον κινοῦν ἀκίνητον), which leaves the created world, man.
The symmetry between the discourse participants disappear preferred position, which belongs to the first initiator, it responds to all participants - including his "license" for completion of the sequence is transferred to all participants, everyone is in possession of "Endlizenz. Thus the question of termination is determined by a general method that anyone applying the rules the general concept of number, can - because he knows the concept of number. Beginning and the end has become something the human world is a part, it is from heaven to earth 'resettled' - by the man himself, in the discourse of social symmetry realized ethics of equality. is
In fact, the construction the number of signs and their generalization is not in numerical concepts mathematics. But it is part of the meta-mathematics (http://de.wikipedia.org/wiki/Metamathematik), which seeks to develop the foundations of mathematics in a consistent theory that escapes the dilemma pointed out by Gödel.
I have shown that the preparatory work for the meta-mathematics, you should actually "Proto Mathematics" call, based on an ethical basis: equal application of the design requirement for all participants in the discourse, and also the additional self-generalization of the participants through the mutual allocation of a symmetrical position the discourse history.
This process in its Preparation of the position of equality is an application of the ethics of Adam Smith (Theory of Moral Sentiments). His "Sympathy" is the ability to symmetrical position exchange, which continues through the "neutral third party" in the transitivity, or the "neutral third party" to the representatives of all are people can - because the nature of his involvement in the discourse of history with each other people can perform well.
We can see the fact that modern science can only be justified if we consider the ethics of social equality, including the explicit recognition of this social Equal rights as basis. Only on this basis, the scientific ethical universals be justified. The expert-based "faith basis" that is connected with the axiomatic method is obsolete. The proto-mathematics is a case that shows the way for other proto-scientific justifications in all subject areas
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KARL JASPERS: The responsibility of freedom
