Disputed term/author/ism  Author 
Entry 
Reference 

Interpretation  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 I 25 BenacerraffVsPlatonism: locus classicus  VsBenacerraf: based on an outdated causal theory of knowledge.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 Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Mathematics  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 antiplatonic 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 "<". (lessthan 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 nonstandard model of such a theory. And that would also apply to quantities. 
Bena I P. Benacerraf Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984 Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Realism  Stalnaker  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. 
Stalnaker I R. Stalnaker Ways a World may be Oxford New York 2003 
Disputed term/author/ism  Author Vs Author 
Entry 
Reference 

Benacerraf, P.  Field Vs Benacerraf, P.  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 nonzero. AntiPlatonismVs: 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 mostdiscussed 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 mindindependent 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. 
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Benacerraf, P.  Lewis Vs Benacerraf, P.  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) 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 noncontingent 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 truthconditional 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. 
Lewis I David K. Lewis Die Identität von Körper und Geist Frankfurt 1989 Lewis I (a) David K. Lewis An Argument for the Identity Theory, in: Journal of Philosophy 63 (1966) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (b) David K. Lewis Psychophysical and Theoretical Identifications, in: Australasian Journal of Philosophy 50 (1972) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (c) David K. Lewis Mad Pain and Martian Pain, Readings in Philosophy of Psychology, Vol. 1, Ned Block (ed.) Harvard University Press, 1980 In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis II David K. Lewis "Languages and Language", in: K. Gunderson (Ed.), Minnesota Studies in the Philosophy of Science, Vol. VII, Language, Mind, and Knowledge, Minneapolis 1975, pp. 335 In Handlung, Kommunikation, Bedeutung, Georg Meggle Frankfurt/M. 1979 Lewis IV David K. Lewis Philosophical Papers Bd I New York Oxford 1983 Lewis V David K. Lewis Philosophical Papers Bd II New York Oxford 1986 Lewis VI David K. Lewis Convention. A Philosophical Study, Cambridge/MA 1969 German Edition: Konventionen Berlin 1975 LewisCl Clarence Irving Lewis Collected Papers of Clarence Irving Lewis Stanford 1970 LewisCl I Clarence Irving Lewis Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 Stalnaker I R. Stalnaker Ways a World may be Oxford New York 2003 
Benacerraf, P.  Wright Vs Benacerraf, P.  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 antiplatonist 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 thanrelation). 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 nonstandard 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) 
WrightCr I Crispin Wright Truth and Objectivity, Cambridge 1992 German Edition: Wahrheit und Objektivität Frankfurt 2001 WrightCr II Crispin Wright "LanguageMastery and Sorites Paradox" In Truth and Meaning, G. Evans/J. McDowell Oxford 1976 WrightGH I Georg Henrik von Wright Explanation and Understanding, New York 1971 German Edition: Erklären und Verstehen Hamburg 2008 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Benacerraf, P.  Verschiedene Vs Benacerraf, P.  Field II 327 Fictionalism/Field: (e.g. Wagner, 1982): a possible representation of Benacerraf's phenomenon: numbers are fictitious objects anyway, and while the fiction in which they occur by default tells us that 0 and 1 precede 2, it does not tell us which, if any, which objects are elements of 2! The question of what they are would be the same as what the wolf had for breakfast before eating the grandmother. Solution/Benacerraf: (also Hellman, 1989): to construct arithmetic so that it is not taken at face value. Then it is not really about the numbers 0,1,2... but about arbitrarily chosen progressions (sequences ((s) "numbered sequences of objects")) of distinguished objects. KitcherVsBenacerraf: (Kitcher 1978): this doesn't really help because the problem occurs for sequences as well as for numbers. HellmanVsKitcher: you can reformulate the idea of sequences in 2. level logic without using special objects. 3. Benacerrraf/Hellman/Field: this can be done in a different way, without requiring a "notforfacevalueinterpretation" (or 2. level logic): you can simply treat mathematics as "referentially indefinite" ((s) >Quine). Our Singular Terms: "0", "1", etc. pretend to pick out certain objects, but do not really do so: so do the general terms: "natural number" 
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Benacerraf, P.  Reductionism Vs Benacerraf, P.  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 ZahlWörtern sprechen kann. mögliche Lösung/Field: jemand könnte sagen daß es nicht wichtig ist, daß die ZahlWörter 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 ZahlWörtern 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 PrimPositionen 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). 
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Benacerraf, P.  Stalnaker Vs Benacerraf, P.  I 50 Truthconditional 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. 
Stalnaker I R. Stalnaker Ways a World may be Oxford New York 2003 
Best Explanation  Fraassen Vs Best Explanation  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: Asif 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 contextdependent, but is that pragmatic? I 91 ContextDependent/ContextIndependent/Theory/Science/Fraassen: theories can also be formulated in a contextindependent 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: ContextDependent/Fraassen: are a) Evaluations of theories, in particular, the term "explained" (explanation) is radically contextdependent. b) the language of the utilization (use) of theories to explain phenomena is radically contextdependent. 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 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Indispensability  Field Vs Indispensability  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: 25269). 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, 1367). ... + ... 
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Platonism  Benacerraf Vs Platonism  Field II 324 BenacerrafVsPlatonism/Field: standard argument: if there are objects as Platonism accepts them, how should we have an epistemic access to them? (Benacerraf 1973). Benacerraf/Field: used an argument against the causal theory of knowledge at the time. PlatonismVsBenacerraf: therefore attacked causal theory. Field: but Benacerraf's objection goes much deeper and is independent of causal theory. Benacerraf: Thesis: a theory can be rejected if it is dependent on the assumption of a massive chance. For example the two statements:´ II 325 (1) John and Judy met every Sunday afternoon last year at different places by chance, (2) they have no interest in each other and would never plan to meet, nor is there any other hypothesis for explanation. ad (2): should make an explanation by some "correlation" impossible. Even if (1) and (2) do not contradict each other directly, they are in strong tension with each other. A belief system that represents both would be highly suspicious. N.B.: but then Platonism is also highly suspicious! Because it postulates an explanation for the correlation between our mathematical beliefs and mathematical facts. (>Access, > Accessibility) For example, why do we only tend to believe that p, if p (for a mathematical p). And for this we must in turn postulate a mysterious causal relationship between belief and mathematical objects. PlatonismVsVs/Field: can claim that there are strong logical connections between our mathematical beliefs. And in fact, in modern times, we can say that we a) tend to conclude reliably and that the existence of mathematical objects serves that purpose; or b) that we accept p as an axiom only if p. FieldVsPlatonism: but this explains reliability again only by some nonnatural mental forces. VsBenacerraf/Field: 1. he "proves too much": if his argument were valid, it would undermine all a priori knowledge (VsKant). And in particular undermine logical knowledge. ("Proves too much"). BenacerrafVsVs/FieldVsVs: Solution: there is a fundamental separation between logical and mathematical cases. Moreover, "metaphysical necessity" of mathematics cannot be used to block Benacerraf's argument. FieldVsBenacerraf: although his argument is convincing VsPlatonism, it does not seem to be convincing VsBalaguer. II 326 BenacerrafVsPlatonismus/Field: (Benacerraf 1965): other approach, (influential argument): 1. For example, there are several ways to reduce the natural numbers to sets: Def natural numbers/Zermelo/Benacerraf/Field: 0 is the empty set and each natural number >0 is the set that contains the set that is n1 as the only element. Def natural numbers/von Neumann/Benacerraf/Field: every natural number n is the set that has as elements the sets that are the predecessors of n. Fact/Nonfactualism/Field: it is clear that there is no fact about whether Zermelo's or Neumann's approach "presents things correctly". There is no fact that determines whether numbers are sets. That is what I call the Def Structuralist Insight/Terminology/Field: Thesis: it makes no difference what the objects of a given mathematical theory are, as long as they are in the right relations to each other. I.e. there is no reasonable choice between isomorphic models of a mathematical theory. …+… 
Bena I P. Benacerraf Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
Wright, Cr.  Field Vs Wright, Cr.  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 antiplatonist 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 "‹" (lessthan 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 ωsequences. I.e. everything that is determined by the use would be a nonstandard 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). 
Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 55367 In Theories of Truth, Paul Horwich Aldershot 1994 
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. 
Stalnaker 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. 
Stalnaker I R. Stalnaker Ways a World may be Oxford New York 2003 
Disputed term/author/ism  Author 
Entry 
Reference 

Truth Conditional Sem.  Stalnaker, R.  I 50 Truth Conditional Semantics/Stalnaker: should be distinguished from a mere division of sentences into two classes, of which one is called "true". Thesis: in order to do this, one should focus on the practice of assertion (assertion), not on an explanation of reference. Assertion/Stalnaker: is more than trying to call a sentence true. Truth ValueAttribution/Stalnaker: is not sufficient to explain assertion and speech acts. We also need a concept of content. The truth value attribution does not tell us why we should claim something, or what a claim could do. Content/Stalnaker: is more than ascribing a truth value. It is also information that can be used for communication. Content/StalnakerVsBenacerraf: the formal counting of horseshoes is not sufficient to attribute content. 
