Lexicon of Arguments


Philosophical and Scientific Issues in Dispute
 


 

Find counter arguments by entering NameVs… or …VsName.

The author or concept searched is found in the following 3 entries.
Disputed term/author/ism Author
Entry
Reference
Interpretation Benacerraf
 
Books on Amazon:
Paul Benacerraf
Field I 22
Interpretation/Benacerraf: (1965) Thesis: Identification of mathematical objects with others is arbitrary - E.g. numbers with quantities. - E.g. real numbers with Dedekind cuts, Cauchy sequences, etc. - There is no fact that decides which is the right one. - Field ditto.
I 22
Indeterminacy of reference/Field: is not a problem, but commonplace.
I 25
For Benacerraf it is about identity, not about reference - otherwise he might falsely be refuted with primitive reference: "Numbers" refers to numbers but not to quantities - But that is irrelevant.
I 25
BenacerraffVsPlatonism: locus classicus - VsBenacerraf: based on an outdated causal theory of knowledge.
Field I 25
BenacerrafVsPlatonism: (1973): if without localization and interaction we cannot know whether they exist. VsBenacerraf: indispensability argument.

Bena I
P. Benacerraf
Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984


Fie I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Fie II
H. Field
Truth and the Absence of Fact Oxford New York 2001

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Mathematics Benacerraf
 
Books on Amazon:
Paul Benacerraf
Field I 20
Mathematics/Identification/Interpretation/Benacerraf: (1965) Thesis: There is an abundance of arbitrariness in the identification of mathematical objects with other mathematical objects: E.g. numbers: numbers can be identified with quantities, but with which?
Real numbers: for them, however, there is no uniform set theoretical explanation. You can identify them with Dedekind's cuts, with Cauchy's episodes,...
---
I 21
...with ordered pairs, with the tensor product of two vector spaces, or with tangent vectors at one point of a manifold. Fact: there does not seem to be a fact that decides which identification to choose! (> Nonfactualism).
Field: the problem goes even deeper: it is then arbitrary what one chooses as fundamental objects, e.g. amounts?
---
Field I 21
Basis/Mathematics/Benacerraf: one can assume functions as fundamental and define sets as specific functions, or relations as basic building blocks and sets as a relation of additivity 1. (adicity). ---
I 23
Mathematics/Indeterminateness/Arbitrariness/Crispin Wright: (1983): Benacerraf's Paper creates no special problem for mathematics: Benacerraf: "Nothing in our use of numerical singular terms is sufficient to specify which, if any amounts are they.
WrightVsBenacerraf: this also applies to the singular terms, which stand for the quantities themselves! And according to Quine also for the singular terms, which stand for rabbits!
FieldVsWright: this misses Benacerraf's argument. It is more against an anti-platonic argument: that we should be skeptical about numbers, because if we assume that they do not exist, then it seems impossible to explain how we have to refer to them or how we have beliefs about them.
According to Benacerraf's argument, our practice is sufficient to ensure that the entities to which we apply the word "number" forms a  sequence of distinct objects under the relation we call "<". (less-than relation). But that's all. Perhaps, however, our use does not even determine this.
Perhaps they only form a sequence that fulfills our best axiomatic theory of the first level of  sequences. That is, everything determined by the use would then be a non-standard model of such a theory. And that would also apply to quantities.

Bena I
P. Benacerraf
Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984


Fie I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Fie II
H. Field
Truth and the Absence of Fact Oxford New York 2001

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Realism Stalnaker
 
Books on Amazon
I 41
Modal Realism/Stalnaker: (thesis that there are possible worlds) - Vsmodal realism: objection: it is not possible, to know any metaphysical facts about it - (whether possible world exist) - thesis: there is no strategy to counter this objection that would be analog to VsBenacerraf. Benacerraf: tension between the need for a plausible representation of mathematical statements and the representation of our respective knowledge about their truth.
I 42
Platonism: gives plausible semantics but no epistemology. - Reference/Benacerraf: thesis: needs causal link. - LewisVsBenacerraf: does not apply to abstract objects such as numbers and so on.
I 47
Conclusion: we cannot distinguish Platonism in terms of mathematical objects from that in terms of possible worlds.
I 49
Modal Realism/VsMR/Possible world/Stalnaker: Problem: the MR cannot say on the one hand that possible worlds are things of the same kind as the real world (contingent physical objects) and on the other hand, that possible worlds are things of which we know in the same way as of numbers, etc. - MR: will insist on the fact that even the reference to ordinary objects (actual or merely possible) needs no causal connection.

Sta I
R. Stalnaker
Ways a World may be Oxford New York 2003


The author or concept searched is found in the following 10 controversies.
Disputed term/author/ism Author Vs Author
Entry
Reference
Benacerraf, P. Field Vs Benacerraf, P.
 
Books on Amazon
I 24
VsBenacerraf/Field: another argument could be brought forward: the problem of consistent arbitrariness of identifications is a phenomenon not only in mathematics, but also in other areas: E.g. PutnamVsMetaphysical Realism: E.g. some say it is arbitrary whether a point is a convergent number of ever smaller regions, all of which are non-zero. Anti-PlatonismVs: If no sets are assumed, the problem takes care of itself.
I 25
Arbitrariness/Field: Thesis: In the realm of physical objects, we do not have the same consistent arbitrariness as in mathematics. VsPlatonism/Mathematics/Field: 1) The most-discussed challenge to him is the epistemological position. Locus classicus: BenacerrafVsPlatonism: (1973): FieldVsBenacerraf: Problem: it relies on an outdated causal theory of knowledge. BenacerrafVsPlatonism: if there were language and mind-independent mathematical entities without spatiotemporal localization which cannot enter any physical interactions, then we cannot know if they exist nor know anything else about them. The Platonist had to postulate mysterious forces. VsBenacerraf: here we could respond with the indispensability argument: Mathematical entities (ME) are indispensable in our different theories about physical objects. FieldVsVs: but this assumes that they are indispensable, and I don’t believe they are. Benacerraf/Field: However, we can formulate his argument more sharply. Cannot be explained as a problem of our ability to justify belief in mathematical entities, but rather the reliability of our belief. In that, we assume that there are positive reasons to believe in such mathematical entities.
I 26
Benacerraf’s challenge is that we need to provide access to the mechanisms that explain how our beliefs about such remote entities reproduces facts about them so well. Important argument: if you cannot explain that in principle, the belief in the mathematical entities wanes. Benacerraf shows that the cost of an assumption of ME is high. Perhaps they are not indispensable after all? (At least this is how I ​​I understand Benacerraf).
I 27
VsBenacerraf/Field: 2) sometimes it is objected to his position (as I have explained) that a declaration of reliability is required if these facts are contingent, which would be dropped in the case of necessary facts. (FieldVs: see below, Essay 7).
I 29
Indispensability Argument/Field: could even be explained with evolutionary theory: that the evolutionary pressure led us to finally find the empirically indispensable mathematical assumptions plausible. FieldVsVsBenacerraf: Problem: the level of mathematics which applied in empirical science is relatively small! That means only this small part could be confirmed as reliable by this empiricism.

Fie I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Fie II
H. Field
Truth and the Absence of Fact Oxford New York 2001

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Benacerraf, P. Lewis Vs Benacerraf, P.
 
Books on Amazon
Field I 231
Example (2) if most mathematicians accept "p" as an axiom, then p.
I 232
VsPlatonism: he has a problem if he cannot explain (2). This is a reformulation of the famous problem of Benacerraf in "Mathematical truth". (see above). (>Benacerraf here departs from a causal theory of truth).
Field: our current approach does not depend on that, though.
I 233
Knowledge/Mathematics/Field: our approach does not depend on the givenness of necessary and sufficient conditions for knowledge. Instead: Reliability Theory/Knowledge/Field: the view that we should be skeptical if the reliability of our knowledge is not explainable in principle.
Mathematics/LewisVsBenacerraf: (Lewis, 1986, p.111 12): Benacerraf's case is not a problem for mathematics because most mathematical facts necessarily apply.
Reliability Theory/Lewis: then we also need an explanation of the reliable relationship, e.g., between facts about electrons and our "electron" belief states and we even have them! In this case, it is the causal approach, according to which the "electron" beliefs counterfactually (>counterfactual conditionals, counterfactual conditional) depend on the existence and nature of electrons.
Explanation/Lewis: now it's precisely the contingent existence and nature of electrons, which makes the question of their existence and nature meaningful.
Lewis: nothing can counterfactually depend on non-contingent things. E.g. nothing can counterfactually depend on which mathematical entities there are. Nothing meaningful can be said about which of our opinions would be different if the number 17 did not exist.
Stalnaker I 41
Mathematics/Benacerraf/Stalnaker: for mathematics we should expect a semantics that is a continuation of general semantics. We should interpret existence statements about numbers, functions and sets with the same truth-conditional semantics as propositions about tables, quarks, etc.
I 42
Knowledge/Mathematics/Reality/Stalnaker: On the other hand, we should also expect that the access to our mathematical knowledge is continuous to the to everyday knowledge. The procedures by which we evaluate and justify mathematical statements should be explained by a general approach to knowledge, together with a representation of mathematical knowledge. Platonism/Mathematics/Benacerraf: Thesis: he gives natural semantics, but does not allow plausible epistemology. ((s) that does not explain how we come to knowledge).
Combinatorial Approach/Combinatorial/Terminology/Benacerraf: Example conventionalism, example formalism: they show mathematical procedures, but do not tell us what the corresponding confirmed mathematical statements tell us.
Benacerraf/Stalnaker: he himself does not offer any solution.
Reference/Benacerraf: Thesis: true reference needs a causal link.
Knowledge/Possible Worlds/Poss.W./Solution/LewisVsBenacerraf: pro Platonism but Vs causal link for reference.

LW I
D. Lewis
Die Identität von Körper und Geist Frankfurt 1989

LW II
D. Lewis
Konventionen Berlin 1975

LW IV
D. Lewis
Philosophical Papers Bd I New York Oxford 1983

LW V
D. Lewis
Philosophical Papers Bd II New York Oxford 1986

LwCl I
Cl. I. Lewis
Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991

Fie III
H. Field
Science without numbers Princeton New Jersey 1980

Sta I
R. Stalnaker
Ways a World may be Oxford New York 2003
Benacerraf, P. Wright Vs Benacerraf, P.
 
Books on Amazon
Field I 23
Mathematics/Uncertainty/Arbitrariness/Crispin Wright (1983): Benacerraf's paper does not create a particular problem for mathematics: Benacerraf: "nothing in our use of numerical singular terms is sufficient to specify which, if any, sets they are.
WrightVsBenacerraf: this also applies for the singular terms representing sets themselves! And according to Quine also for the singular terms that stand for rabbits!
FieldVsWright: this goes past Benacerraf's argument. It is aimed more against an anti-platonist argument: that we should be skeptical about numbers, because if we assume that they do not exist, then it seems to be impossible to explain how we refer to them or have beliefs about them.
According to Benacerraf's argument our practice is sufficient to ensure that the entities to which we apply the word "number" form a sequence of distinct objects, under the relation that we call "<". (less than-relation). But that's all. But perhaps our use not does not even determine that.

             Perhaps they only form a sequence that satisfies our best axiomatic theory of the first stage of w sequences. I.e. everything that is determined by use, would be a non-standard model of such a theory. And that would also apply to sets.

             Wright (s): Thesis our standard use is not sufficient for the determination of the mathematical entities. (FieldVsWright).

             I 24
             VsWright: but the assertion that this also applied to rabbits is more controversial. A bad argument against this would be a causal theory of knowledge (through perception)

Wri I
Cr. Wright
Wahrheit und Objektivität Frankfurt 2001

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Benacerraf, P. Verschiedene Vs Benacerraf, P. Field II 327
Fiktionalismus/Field: (Bsp Wagner, 1982): eine mögliche Darstellung des Phänomens von Benacerraf: Zahlen sind sowieso fiktive Objekte, und während die Fiktion, in der sie standardmäßig vorkommen, erzählt, daß 0 und 1 der 2 vorausgehen, sagt sie uns nicht, welche, wenn überhaupt welche Objekte Elemente von 2 sind! Die Frage, welche es sind, käme der Frage gleich, Bsp was der Wolf zum Frühstück hatte, bevor er die Großmutter fraß. Lösung/Benacerraf: (auch Hellman, 1989): Arithmetik so zu konstruieren, daß sie nicht für bare Münze genommen wird. Sie handelt dann nicht wirklich von den Zahlen 0,1,2... sondern von willkürlich gewählten Progressionen (-Sequenzen ((s) „numerierten Folgen von Gegenständen“) ) von unterschiedenen Objekten.
KitcherVsBenacerraf: (Kitcher 1978): das hilft nicht wirklich, weil das Problem für -Sequenzen genauso auftritt wie für Zahlen.
HellmanVsKitcher: man kann die Idee der  -Sequenzen in Logik 2. Stufe reformulieren, ohne den Gebrauch spezieller Objekte.
3.
Benacerrraf/Hellman/Field: das kann man noch anders durchführen, ohne eine „nicht-für-bare-Münze-Interpretation“ (oder Logik 2. Stufe) zu verlangen: man kann Mathematik einfach als „referentiell unbestimmt“ ((s) >Quine) behandeln. Unsere
sing Term: „0“, „1” usw. geben vor, bestimmte Objekte herauszugreifen, tun es aber nicht wirklich: ebenso die
allg Term: „natürliche Zahl“, „





Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Benacerraf, P. Reductionism Vs Benacerraf, P.
 
Books on Amazon
Field II 214
Reduktion/Denotation/BenacerrafVsReduktion/Field: (Benacerraf, 1965): Problem: hier kann es mehrere Korrelationen geben, so daß man unmöglich von dem „wirklichen Referenten“ von Zahl-Wörtern sprechen kann. mögliche Lösung/Field: jemand könnte sagen daß es nicht wichtig ist, daß die Zahl-Wörter gerade auf diese Objekte referiert, es ist hinreichend (könnte er sagen), daß wir die Rede über Zahlen durch die Rede über Objekte ersetzen können. (Quine 1960. § § 53,54).
FieldVsQuine: das würde die Lehrsätze von Euler und Gauß zu Sätzen erklären, die mit ihren Zahl-Wörtern auf nichts referieren und letztlich falsch wären.
Benacerraf/Field: scheint damit jede Reduktion auszuschließen.
ReduktionismusVsBenacerraf/Field: Autoren, die glauben, daß es abstrakte Gegenstände gibt, die keine Mengen sind, (d.h. Zahlen) könnten sagen: alles was Benacerraf damit zeigt, daß es eine eineindeutige Relation gibt. Zur Reduktion braucht man aber nur eine Erklärung zahlentheoretischer Wahrheit in Begriffen einer Korrespondenz zwischen Zahlwörtern auf der einen Seite und physischen Objekten und/oder Mengen auf der anderen Seite. (Mit einer Verallgemeinerung gilt das auch für Gavagai).
II 214/215
Bsp „prim“: relativ zu jeder  -Sequenz s die mit den Zahlen korreliert ist, signifiziert „prim“ ((s) nicht partiell!) die Prim-Positionen von s. Pointe: dann ist ein Satz wie Bsp „Die Zahl zwei ist Cäsar“ weder wahr noch falsch (OWW).
FieldVsBenacerraf: seine Beobachtung ist also umgehbar. Wir können mathematische Wahrheit bewahren. (>Wahrheitserhalt).

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Benacerraf, P. Stalnaker Vs Benacerraf, P.
 
Books on Amazon
I 50
Truth-conditional semantics/Stalnaker: should one differ from a mere classification of propositions into two classes of one which is called a "true". Thesis: to do that one should concentrate on the practice of asserting (assertion) concentrate not on an explanation of the reference.
Assertion/Stalnaker: is more than to try to call a proposition true.
Ascription of truth values/Stalnaker: is not sufficient to explain assertion and speech acts. We also need a concept of content. The ascription of truth values does not tell us why we should say something or what an assertion could cause.
Content/Stalnaker: is more than ascription of a truth value. It is also an information that can be used for communication.
Content/StalnakerVsBenacerraf: the formal counting of horseshoes is not sufficient for an ascription of content.
Proposition/Stalnaker: may also be contingent.

Sta I
R. Stalnaker
Ways a World may be Oxford New York 2003
Best Explanation Fraassen Vs Best Explanation
 
Books on Amazon
Field I 15
Principle of the Best Explanation/Field: Suppose we have a) certain beliefs about the "phenomena" that we do not want to give up
b) this class of phenomena is large and complex
c) we have a pretty good (simple) explanation that is not ad hoc and from which the consequences of the phenomena follow
d) one of the assumptions in the explanation is assertion S and we are sure that no explanation is possible without S.
Best Explanation: then we have a strong reason to believe S.
False: "The phenomena are as they would be if explanation E was correct":
As If/Field: As-if assertions that are piggyback passengers on true explanations may not be constructed as explanations themselves (at least not ad hoc).
Then the principle is not empty: it excludes the possibility that we accept a large and complex set of phenomena as a brute fact.
(van FraassenVsBest Explanation: 1980)
Best Explanation/BE/Field: the best explanation often leads us to believe something that we could also test independently by observation, but also to beliefs about unobservable things, or unobservable beliefs about observable things.
Observation: should not make a difference here! In any case, our beliefs go beyond what is observed.
I 16
Important argument: if no test was done, it should make no difference in the status of the evidence between cases where an observation is possible and those where no observation is possible! A stronger principle of the best explanation could be limited to observable instances of belief.
FieldVs: but that would cripple our beliefs about observable things and would be entirely ad hoc.
Unobserved things: a principle could be formulated that allowed the inference on observed things - that have been unobserved so far! - while we do not believe the explanation as such.
FieldVs: that would be even more ad hoc!
I 25
VsBenacerraf: bases himself on an outdated causal theory of knowledge. - - -
I 90
Theory/Properties/Fraassen: theories have three types of properties: 1) purely internal, logical: axiomatization, consistency, various kinds of completeness.
Problem: It was not possible to accommodate simplicity here. Some authors have suggested that simple theories are more likely to be true.
FraassenVsSimplicity: it is absurd to suppose that the world is more likely to be simple than that it was complicated. But that is metaphysics.
2) Semantic Properties: and relations: concern the relation of theory to the world. Or to the facts in the world about which the theory is. Main Properties: truth and empirical adequacy.
3) pragmatic: are there any that are philosophically relevant? Of course, the language of science is context-dependent, but is that pragmatic?
I 91
Context-Dependent/Context-Independent/Theory/Science/Fraassen: theories can also be formulated in a context-independent language, what Quine calls Def "External Sentence"/Quine. Therefore it seems as though we do not need pragmatics to interpret science. Vs: this may be applicable to theories, but not to other parts of scientific activity:
Context-Dependent/Fraassen: are
a) Evaluations of theories, in particular, the term "explained" (explanation) is radically context-dependent.
b) the language of the utilization (use) of theories to explain phenomena is radically context-dependent.
Difference:
a) asserting that Newton’s theory explains the tides ((s) mention).
b) explaining the tides with Newton’s theory (use). Here we do not use the word "explains".
Pragmatic: is also the immersion in a theoretical world view, in science. Basic components: speaker, listener, syntactic unit (sentence or set of sentences), circumstances.
Important argument: In this case, there may be a tacit understanding to let yourself be guided when making inferences by something that goes beyond mere logic.
I 92
Stalnaker/Terminology: he calls this tacit understanding a "pragmatic presupposition". (FraassenVsExplanation as a Superior Goal).
I 197
Reality/Correspondence/Current/Real/Modal/Fraassen: Do comply the substructures of phase spaces or result sequences in probability spaces with something that happens in a real, but not actual, situation? ((s) distinction reality/actuality?) Fraassen: it may be unfair to formulate it like that. Some philosophical positions still affirm it.
Modality/Metaphysics/Fraassen: pro modality (modal interpretation of frequency), but that does not set me down on a metaphysical position. FraassenVsMetaphysics.
I 23
Explanatory Power/Criterion/Theory/Fraassen: how good a choice is explanatory power as a criterion for selecting a theory? In any case, it is a criterion at all. Fraassen: Thesis: the unlimited demand for explanation leads to the inevitable demand for hidden variables. (VsReichenbach/VsSmart/VsSalmon/VsSellars).
Science/Explanation/Sellars/Smart/Salmon/Reichenbach: Thesis: it is incomplete as long as any regularity remains unexplained (FraassenVs).

Fr I
B. van Fraassen
The Scientific Image Oxford 1980

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Indispensability Field Vs Indispensability
 
Books on Amazon
I 14
Indispensability Argument/Field: here it’s all about purposes - such an argument must be based on the best explanation (BE).
I 17
FieldVsIndispensability Argument: we can show that there are good theories that do without mathematical entities - Justification/Field: is gradual. FieldVsIndispensability Argument: two points which together make it seem untenable: 1) if we can show that there are equally good theories that do not involve ME. I believe that we can show that in the case of ME, but not in the case of electrons! (Lit.Field: "Science without Numbers"). At the moment, we do not yet know exactly how to eliminate ME, and our method of ((s) complete) induction gives us some confidence in mathematical entities 2) Justification is not a question of all or nothing! (justification gradual)
I 29
Indispensability Argument/Field: Might even be explained by way of evolutionary theory: that evolutionary pressure finally led us to find the empirically indispensable mathematical assumptions plausible. FieldVsVsBenacerraf:. Problem: the scope of mathematics which is used in empirical science is relatively small! That means that only this small portion could be confirmed as reliable by empiricism. And inferences on the rest of mathematics are not sustainable, there are simply too many possible answers to questions about large cardinals or the continuum hypothesis or even about the axiom of choice. These work well enough to provide us with the simpler "application mathematics". ((s) That means that we cannot infer a specific answer to the questions of the higher levels from application mathematics.)
II 328
Utility/Truth/Mathematics/Putnam/Field: (Putnam 1971 locus classicus, unlike 1980): Thesis: we must consider mathematics as true in order to be able to explain its utility (benefit) in other fields. E.g. in science and metalogic. (i.e. the theory of logical consequence). Modality/Modal/Mathematics/Field: this is in contrast to his former view that we can use modality instead of mathematical objects to explain mathematical truth.
II 329
Modal Explanation: will not work for other disciplines such as physics, however. (FieldVsPutnam, Field 1989/91: 252-69). Putnam/Field: the general form of his argument is this: (i) we must speak in terms of mathematical entities in order to study science, metalogic, etc. (ii) If need them for such important purposes, we have reason to believe that this kind of entities exists. VsPutnam/Field: there are two possible strategies against this: 1) Vs: "foolhardy" strategy: requires us to substantially change premise (i): we want to show that we basically do not need to make any assumptions which require mathematical entities. I.e. we could study physics and metalogic "nominalistically". Problem: in a practical sense, we still need the mathematical entities for physics and metalogic. We need to explain this practical indispensability. "foolhardy" strategyVs: in order to explain them, we just have to show that mathematical entities are only intended to facilitate inferences between nominalistic premises. And if this facilitation of inference is the only role of mathematical entities, then (ii) fails. Solution: In that case, something much weaker than truth (E.g. "conservatism") suffices as an explanation for this limited kind of utility. FieldVs: Unfortunately, the project of nominalization is not trivial. (Field 1980 for physics, 1991 for metalogic). At that time I found only few followers, but I am too stubborn to admit defeat. 2) Vs ("less foolhardy strategy"): questions (ii) more profoundly: it denies that we can move from the theoretical indispensability of existence assumptions to a rational belief in their truth. That is what Putnam calls "indispensability argument". Putnam pro. FieldVsPutnam: that requires some restrictions and ManyVsPutnam: these restrictions ultimately prevent an application in mathematics. And ultimately, because mathematical entities are simply not causally involved in physical effects.
II 330
FieldVsPutnam: that’s plausible. PutnamVsVs: If mathematical entities are theoretically indispensable in causal explanations (such as (i) claims), however, there seems to be a sense in which they are very well causally involved. Conversely, it would have to be explained why they should not be causally involved. FieldVs: a closer look should reveal that the role of mathematical entities is not causal. And that it supports no indispensability argument. E.g. the role of quantities in physics was simply to allow us to assert the local compactness of physical space. Other E.g. role of quantities in physics. Allow us to accept (Cp) instead of (Cs). (Field, 1989) 1, 136-7). ... + ...

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Platonism Benacerraf Vs Platonism
 
Books on Amazon:
Paul Benacerraf
Field II 324
BenacerrafVsPlatonismus/Field: Standardargument: wenn es Objekte gibt so wie der Platonismus sie annimmt, wie sollten wir einen epistemischen Zugang zu ihnen haben? (Benacerraf 1973). Benacerraf/Field: gebrauchte damals ein Argument gegen die Kausaltheorie des Wissens.
PlatonismusVsBenacerraf: griff daher die Kausaltheorie an.
Field: aber Benacerrafs Einwand geht viel tiefer und ist von der Kausaltheorie unabhängig.
Benacerraf: These: eine Theorie kann zurückgewiesen werden, wenn sie von der Annahme eines massiven Zufalls abhängig ist. Bsp die zwei Aussagen:
II 325
(1) John und Judy haben sich jeden Sonntag nachmittag im letzten Jahr zufällig an verschiedenen Orten getroffen, (2) sie haben kein Interesse aneinander und würden nie planen sich zu treffen, auch gibt es keine andere Hypothese zur Erklärung.
ad (2): soll eine Erklärung durch irgendeine „Korrelation“ unmöglich machen.
Wenn (1) und (2) sich auch nicht direkt widersprechen, stehen sie doch in starker Spannung zueinander. Ein Glaubenssystem, das beide vertritt, wäre höchst verdächtig.
Pointe: dann ist aber auch der Platonismus höchst verdächtig! Denn er postuliert eine Erklärung für die Korrelation zwischen unseren mathematischen Glaubenseinstellungen und mathematischen Tatsachen. (>Zugang, > Zugänglichkeit) Bsp warum wir nur dann dazu tendieren zu glauben, dass p, wenn p (für ein mathematisches p). Und dafür müssen wir wiederum einen mysteriösen kausalen Zusammenhang postulieren, zwischen Glauben und mathematischen Objekten.
PlatonismusVsVs/Field: kann sich darauf berufen, dass es starke logische Verbindungen zwischen unseren mathematischen Überzeugungen gibt. Und in der Tat, in der modernen Zeit kann man sagen dass wir
a) dazu tendieren, verläßlich zu schließen, und dass die Existenz mathematischer Objekte dem dienen oder
b) dass wir p als Axiom nur akzeptieren, wenn p.
FieldVsPlatonismus: das erklärt aber die Verläßlichkeit wieder nur durch irgendwelche nicht- natürlichen geistigen Kräfte.
VsBenacerraf/Field: 1. er „beweist zu viel“: wenn sein Argument gültig wäre, würde es alles a priori Wissen unterminieren (VsKant). Und insbesondere logisches Wissen unterminieren. („Beweist zu viel“).
BenacerrafVsVs/FieldVsVs: Lösung: es gibt eine fundamentale Trennung zwischen logischen und mathematischen Fällen. Außerdem kann man „metaphysische Notwendigkeit“ der Mathematik nicht dazu gebrauchen, Benacerrafs Argument zu blockieren.
FieldVsBenacerraf: obwohl sein Argument überzeugen VsPlatonismus ist, scheint es nicht überzeugend VsBalaguer zu sein. II 326
BenacerrafVsPlatonismus/Field: (Benacerraf 1965): anderer Ansatz, (einflußreiches Argument):
1.
Bsp es gibt verschiedene Möglichkeiten, die natürlichen Zahlen auf Mengen zu reduzieren: Def natürliche Zahlen/Zermelo/Benacerraf/Field: 0 ist die leere Menge und jede natürliche Zahl >0 ist die Menge, die als einziges Element die Menge die n-1 ist, enthält.
Def natürliche Zahlen/von Neumann/Benacerraf/Field: jede natürliche Zahl n ist die Menge, die als Elemente die Mengen hat, die die Vorgänger von n sind.
Tatsache/Nonfaktualismus/Field: es ist klar, dass es keine Tatsache darüber gibt, ob Zermelos oder von Neumanns Ansatz die Dinge „richtig darstellt“. Es gibt keine Tatsache die entscheidet, ob Zahlen Mengen sind.
Das nenne ich die
Def strukturalistische Einsicht/Terminologie/Field: These: es macht keinen Unterschied, was die Objekte einer gegebenen mathematischen Theorie sind, so lange sie in den richtigen Relationen zueinander stehen. D.h. es gibt keine sinnvolle Wahl zwischen isomorphen Modellen einer mathematischen Theorie. …+…

Bena I
P. Benacerraf
Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984

Fie III
H. Field
Science without numbers Princeton New Jersey 1980
Wright, Cr. Field Vs Wright, Cr.
 
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I 23
Mathematics/Indeterminacytainty/Arbitrariness/Crispin Wright: (1983): Benacerraf’s paper creates indeterminacy not a particular problem for mathematics: Benacerraf: "nothing in our use of numerical singular terms is sufficient to specify which, if any, quantities they are.
WrightVsBenacerraf: this is also valid for the singular terms that represent the quantities themselves! And according to Quine also for singular terms that stand for rabbits!
FieldVsWright: this goes past Benacerraf’s argument. It is aimed more against an anti-platonist argument: that we should be skeptical of numbers, because if we assume that they do not exist, then it seems to be impossible to explain how we refer to them or have beliefs about them.
According to Benacerraf’s argument our practice is sufficient to ensure that the entities to which we apply the word "number" form a  sequence of distinct objects under the relation that we call "‹" (less-than relation). But that’s all. But perhaps our use does not even determine this. Perhaps they only form a sequence that fulfills our best axiomatic first level theory of ω-sequences. I.e. everything that is determined by the use would be a non-standard model of such a theory. And that would then also apply for quantities. Wright/(s): Thesis: Our standard use is not sufficient for determining the mathematical entities. (FieldVsWright). I 24 VsWright: but that this would apply for rabbits is more controversial. A bad argument against it would be a causal theory of knowledge (through perception).

Fie III
H. Field
Science without numbers Princeton New Jersey 1980

The author or concept searched is found in the following disputes of scientific camps.
Disputed term/author/ism Pro/Versus
Entry
Reference
Causal theory of knowledge Versus Stalnaker I 42
Knowledge/causality/causal theory of knowledge: Benacerraf: Thesis: knowledge / reference requires causal connection. - LewisVsBenacerraf: abstract objects such as numbers, etc. do not require a causal connection - Stalnaker ditto.

Sta I
R. Stalnaker
Ways a World may be Oxford New York 2003
Causal theory of knowledge Pro Stalnaker I 42
Knowledge/causality/causal theory of knowledge: Benacerraf: Thesis: knowledge/reference requires causal connection - LewisVsBenacerraf: abstract objects such as numbers, etc. do not require a causal connection - Stalnaker ditto.

Sta I
R. Stalnaker
Ways a World may be Oxford New York 2003

The author or concept searched is found in the following theses of the more related field of specialization.
Disputed term/author/ism Author
Entry
Reference
Truth Conditional Sem. Stalnaker, R.
 
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I 50
Wahrheits-konditionale Semantik/Stalnaker: sollte man von einer bloßen Einteilung von Sätzen in zwei Klassen, von denen man eine -žwahr-œ nennt, unterscheiden. These Um das zu tun, sollte man sich auf die Praxis des Behauptens (Behauptung) konzentrieren, nicht auf eine Erklärung der Referenz.
Behauptung/Stalnaker: ist mehr, als zu versuchen, einen Satz wahr zu nennen.
WW-Zuschreibung/Stalnaker: ist nicht hinreichend, um Behauptung und Sprechakte zu erklären. Wir brauchen auch einen Begriff von Inhalt. Die WW-Zuschreibung sagt uns nicht, warum wir etwas behaupten sollten, oder was eine Behauptung bewirken könnte.
Inhalt/Stalnaker: ist mehr als Zuschreibung eines WW. Er ist auch Information, die zur Kommunikation gebraucht werden kann.
Inhalt/StalnakerVsBenacerraf: das formale Zählen von Hufeisen ist für eine Zuschreibung von Inhalt nicht hinreichend.