Dictionary of Arguments


Philosophical and Scientific Issues in Dispute
 
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The author or concept searched is found in the following 14 controversies.
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 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 Locus classicus: BenacerrafVsPlatonism: (1973): FieldVsBenacerraf: Problem: it relies on an outdated causal theory of knowledge.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.

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. 553-67
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 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.

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. 3-35
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. 553-67
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 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)

WrightCr I
Crispin Wright
Truth and Objectivity, Cambridge 1992
German Edition:
Wahrheit und Objektivität Frankfurt 2001

WrightCr II
Crispin Wright
"Language-Mastery 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. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Deflationism Field Vs Deflationism I 102
Applicability/Mathematics/VsDeflationism: Problem: (see above): Deflationism cannot explain the benefits of the proof theory without having to assume it truth ((s) and thus the existence of mathematical entities (ME). Two attempts at a solution:. 1) there may be a nominalist proof theory which is just as good as the Platonic one. But that would only be a change of subject, as long as nothing more is explained. 2) trying to the explain Platonic proof th. without assuming that it is true.
Field II 126
VsDeflationism/Field:
1) Vs: if one simply accepts the T-sentences, that has nothing to do with the content. ((s) Due to equivalence which ​​requires only equal truth values). So the language is cut off from the world. E.g. "There are gravitational waves" is true iff there are gravitational waves
has, disquotationally seen, nothing to do with gravitational waves. So we should have a connection between our use of the term "gravitational waves" and gravitational waves, regardless of the scheme.
DeflationismVsVs/Field: but Deflationism allows that precisely: it allows for facts that
II 127
Refer sentences on gravitational waves regardless of the disquotational truth. E.g. laws of physics. The use is not the only fact that exists here. 2) VsDeflationism/Field: (most important): it cannot explain the explanatory power of the truth conditions E.g. for explanation of behavior, or the explanation of how far behavior is successful.
3) VsDeflationism/Field: it cannot distinguish between a vague and non-vague discourse or between a discourse which is based on facts and one that is not. The following are less important and are discussed in the following sections.
4) VsDeflationism: it cannot handle truth attributions in other languages.
5) VsDeflationism: it gives "true" false modal properties (s) "necessarily true" or "contingently true").
6) VsDeflationism: it cannot handle ambiguity, indices and demonstratives.
7) VsDeflationism: it cannot explain how we learn from others.
FieldVsVs: 4 - 7 per Deflationism. Here my version of Deflationism is radical.
II 135
Index Words/Demonstratives/Truth Conditions/Deflationism/Field: we must distinguish two stages of sentences that contain them: 1) focuses on sentence types: there must be no unrelativized T predicate here E.g. a sentence type like "I do not like her" has no truth value. Solution: we can associate a truth value corresponding to a pair of objects : then the sentence is true relative to if b dislikes x. Field: this is not "strictly disquotational", because it involves a grammatical change. 2) Then we need access of unrelativized truth for sentence tokens. That means we must assign an object to each index element. I/Now: is no problem here: that’s "the author of the utterance" or "the time of the utterance". But that is not possible with the others. VsDeflationism: for assigning "this" or "he" we need semantic terms, i.e. it does not work in a purely disquotational way.
II 137
Learning/VsDeflationism/Field: Thesis: you need inflationism to explain the learning from others, because we assume that most of what other people tell us is true. ((s) so it is purely disquotational, not merely a repetition, or "true, if the sentence is repeated, because you do not learn meanings from repetition, you need something like paraphrases.). VsDeflationism/Field: 1) with learning some kind of translation must be involved so that a certain inter-personal synonymy is presumed in the inference. 2) even purely disquotational truth + synonymy is not sufficient: E.g. My friend Charley said that in Alabama (a southern state) there was a feet of snow (which never happens).
II 138
VsDeflationism/Solution: the reformulated inference works, because a more substantial property is attributed than merely disquotational truth.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Field, H. Gödel Vs Field, H. Field I 66
Realism/Mathematics/Gödel: ("What is Cantor's Continuum Problem?", 1947) (Pro Quine Putnam Argument, VsField, VsAnti Realism): even with a very narrow definition of the term "mathematical data" (only equations of number theory) we can justify quite abstract parts by explanatory success: Gödel: even without having to accept the necessity of a new axiom, and even if it has no intrinsic necessity at all, a decision about its truth is possible by examining its "explanatory success" with induction.
The fertility of its consequences, especially the "verifiable" ones, i.e. those which can be demonstrated without the new axiom, but whose proofs are made easier by the new axiom. Or if one can combine several proofs to one.
For example the axioms about the real numbers, which are rejected by the intuitionists.
I 67
FieldVsGödel: if no mathematical entities are indispensable, then one does not have to call the so-called "mathematical data" true. But at the beginning I had said that there can be no other goal of mathematics than truth.

Göd II
Kurt Gödel
Collected Works: Volume II: Publications 1938-1974 Oxford 1990

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Field, H. Wright Vs Field, H. Field I 43
Anti-Platonism/AP/WrightVsField: (Hale, 1987): claims that modal considerations undermine my version of the anti-platonism. Because I take mathematics and the existence of mathematical entities (mE) as consistent, and consistency as the modal basic concept (possibility), I would be bound to think that it is wrong that there are mathematical entities - that the existence of mathematical entities is "contingent wrong". ((s) "There could just as well have been mathematical entities, i.e. empirical question").
Contingent/Wright/HaleVsField: is not logical, and thus something other than "neither logically true nor logically contradictory". And that makes Field's position absurd.
WrightVsField: where should Fields "contingency" be contingent on? For example, according to Field, the actual world contains no numbers, but it could have contained some. But there is neither an explanation for why not, nor would there be an explanation if there were numbers.
FieldVsVs: if the argument were good, it would be equally valid against (nonlogical) platonism, for which mathematics goes back behind logic. Then the denial of all mathematics would be logically consistent and therefore "contingent". But this is a confusion of the different meanings of "possible". Analog:
For example, if the existence of God is logically consistent, and there is none, then it is contingent wrong that there would be one.
Problem: the atheist has no access to what the contingent is supposed to be on. There would be neither an explanation for the existence nor for the non-existence. There are no favorable conditions for God's existence and no unfavorable ones. (>Anselm, 2. Ontological argument).
But WrightVsField: has even more interesting arguments: 1. without the assumption that mathematics consists of necessary truths, the view that mathematics is conservative (preserving) is unjustified.
I 44
Analog: without the assumption that mathematics is true, the assumption that it is consistent is unjustified. Justification/FieldVsWright: You can justify any belief by a stronger belief from which it follows. (>Strength of Theories).
Wright and Hale would have to show that Platonism has better reasons for the necessary truth of mathematics than Anti-Platonism has for assuming that mathematics is conservative (or consistent). And it is not certain that this is true.
WrightVsField: 2. Anyone who represents both:
a) that the existence of mathematical entities is "contingent false" and
b) that mathematics is conservative,
can give no reason not to believe in mathematical entities!
Def Conservativity/Mathematics/Field: means that any internally consistent combination of nominalistic statements is also consistent with mathematics.
Then no combination of nominalistic statements can provide an argument against belief in mathematics (ontology).
WrightVsField: how then can there be any reason at all not to believe in mathematics? He has no proof of his own nominalism. It follows that Field cannot be a nominalist, but that he must be an agnostic.
FieldVsWright: this one misjudges the relevance that I attribute to the question of renunciability and indispensability.
Conservativity: does not automatically show that there can be no reason to believe in mathematics.
To succeed with VsPlatonism, we must also show that mathematics is dispensable in science and meta logic. Then we have reason not to literally have to believe in mathematics.
I 45
If that succeeds, we can get behind the agnosticism.

WrightCr I
Crispin Wright
Truth and Objectivity, Cambridge 1992
German Edition:
Wahrheit und Objektivität Frankfurt 2001

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. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Idealism Field Vs Idealism I 27
"Mathematical Idealism"/Field: mathematical entities as "mental constructions". FieldVs: that is obscure. Idealists: the representatives would argue that it is not difficult to explain the reliability of our belief in entities that we ourselves have constructed!
FieldVsIdealism: 1) can extract no sense at all from classical mathematics, if it assumes constructions. (Brouwer and Heyting reacted to this with intuitionism).
2) those who designate the ME as mental constructions make no corresponding statements about physical entities, which makes applicability a mystery.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
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: 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). ... + ...

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Lewis, D. Field Vs Lewis, D. I 233
Knowledge/Belief/Explanation/Mathematics/Lewis: consequently, since mathematics consists of necessary truths, there can be no explanation problem. FieldVsLewis: at least 4 points, why this does not exclude the epistemic concerns:
1) not all the facts about the realm of mathematical antities apply necessarily. But suppose it were so, then there are still facts about the mathematical and non-mathematical realm together! E.g.
(A) 2 = the number of planets closer to the Sun than the Earth.
(B) for a natural number n there is a function that depicts the natural numbers smaller than n on the set of all particles in the universe ((s) = there is a finite number of particles).
(C) beyond all sp.t. points there is an open region, for which there is a 1: 1 differentiable representation.
I 234
of this region on an open subset of R4 (space, quadruples of real numbers). (D) there is a differentiable function y of spatial points on real numbers, so that the gradient of y indicates the gravitational force on each object, as measured by the unit mass of that object.
Field: these facts are all contingent. But they are partly about the mathematical realm (mathematical entities).
Explanation/FieldVsLewis: There remains the problem of the explanation of such "mixed" statements. (Or the correlation of these with our beliefs).
Solution: You can divide these statements: an
a) purely mathematical component (without reference to physical theories, but rather on non-mathematical entities, E.g. quantities with basic elements, otherwise the condition would be too strong). Important argument: this component can then be regarded as "necessarily true".
b) purely non-mathematical component (without reference to mathematics).
I 235
2) FieldVsLewis: even with regard to purely mathematical facts, Lewis’ answer is too simple. Necessary Facts/Mathematics: to what extent should they be necessary in the realm of mathematics? They are not logically necessary! And they cannot be reduced to logical truths by definition.
Of course they are mathematically necessary in the sense that they follow from the laws of mathematics.
E.g. Similarly, the existence of electrons is physically necessary, because it follows from the laws of physics.
FieldVsLewis: but in this physical case, Lewis would not speak of a pseudo-problem! But why should the fact that numbers exist mathematically necessary be a pseudo-problem?.
Mathematical Necessity/Field: false solution: you could try to object that mathematical necessity is absolute necessity, while physical necessity is only a limited necessity.
Metaphysical Necessity/Field: or you could say that mathematical statements.
I 236
Are metaphysically necessary, but physical statements are not. FieldVs: It is impossible to give content to that.
I 237
3) FieldVsLewis: he assumes a controversial relation between Counterfactual Conditional and necessity. It is certainly true that nothing meaningful can be said about E.g. what would be different if the number 17 did not exist. And that is so precisely because the antecedent gives us no indication of what alternative mathematics should be considered to be true in this case.
I 238
4) FieldVsLewis: there is no reason to formulate the problem of the explanation of the reliability of our mathematical belief in modal or counterfactual expressions.
II 197
Theoretical Terms/TT/Introduction/Field: TT are normally not introduced individually, but in a whole package. But that is no problem as long as the correlative indeterminacy is taken into account. One can say that the TT are introduced together as one "atom". E.g. "belief" and "desire" are introduced together.
Assuming both are realized multiply in an organism:
Belief: because of the relations B1 and B2 (between the organism and internal representations).
Desired: because of D1 and D2.
Now, while the pairs (B1, D1) and (B2, D2) have to realize the (term-introductory) theory.
II 198
The pairs (B1, D2) and (B2, D1) do not have to do that. ((s) exchange of belief and desire: the subject believes that something else will fulfill its desire). FieldVsLewis: for this reason we cannot accept its solution.
Partial Denotation/Solution/Field: we take the TT together as the "atom" which denotes partially as a whole.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Modalities Field Vs Modalities I 268
FieldVsModality/Science/Mathematics: (see above): it would be no use nothing if the modal nominalism eliminated the technical difficulty; that would only show that mathematical entities could be eliminated modally in a manner that is no better than the modal elimination of subatomic particles. If modality to be useful, it must be less trivial. I do not think that’s possible.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Possibilia Lewis Vs Possibilia Schwarz I 87
Possibilia/Possible World/possible worlds/possibilistic structuralism/Lewis/Schwarz: (1991(1),1993d(2)) here Lewis assumed, thesis: that there are clearly less inhabitants of possible worlds (Possibilia) than sets. Set theory: so for them additional entities had to be accepted besides the Possibilia. These additional entities should then contain the sets (and classes) required by the 5th condition (see above).
Lewis later: accepts that there are at least as many Possibilia as sets. Then one could do without the additional mathematical entities (Lewis pro). Then we delete condition five. Then "many" inhabitants of possible worlds must be sets.
Schwarz I 88
Because Lewis assumes that there are more sets than individuals. Because if there are "many" individuals, then also "many" individual atoms, atoms of individuals exist. But there are more sums of individual atoms than individual atoms. Then there are also more individuals than atoms at all and then, according to conditions (1) and (3), more units than atoms, in contradiction to (2). Possibilia/Lewis/Schwarz: if they have no cardinality, not all Possibilia can be individuals.
Def possibilistic structuralism/Lewis/Schwarz: mathematical statements are not only about mathematical entities anyway, but partly also about Possibilia. Then why not just these?
Pro: not only does he get along without primitive mathematical vocabulary, but also without primitive mathematical ontology. Questions about their origin and our epistemic approach are thus resolved. If mathematical statements are about Possibilia, it results in a modal state from the logic of unlimited modality: For unlimited modal statements truth, possibility and necessity coincide.(see section 3.6 above).
Lewis: can't just delete the mathematical entities. (LewisVsField): Problem: mixed sums. For example, if some atoms in Caesar's brain are classified as single sets and others as individuals, then Caesar is a mixed sum.
Mixed Sum/Mereology/Lewis: is neither individual nor class.
Class: Sum of single sets.
Schwarz I 89
Mixed sums: are not elements of sets in Lewis' original system either. Schwarz: that is unmotivated in terms of set theory: according to the iterative view, absolutely everything has a single set. Lewis usually ignores mixed sums anyway.
Problem: not under every single set relationship is there a single set of Caesar.
Solution: a) also allow a mixed sums single set. Vs: there are more mixed sums than single sets, so that doesn't work.
b) Requirement: that all "small" mixed sums have a single set.
c) More elegant: settle mixed sums by forbidding individuals. If you identify classes with ordinary Possibilia, you could treat each atom as a single set. For example, Caesar is then always a class, his single set is the object of pure set theory.
LewisVs: this does not work in his set theory (unlike ZFC). Because we need at least one individual as an empty set.
Single set/Lewis/Schwarz: since a single individual atom is sufficient, instead of (1) (3) single set relationships, one could also determine arbitrary unambiguous images of small things in all atoms except one. This one atom is then the empty set relative to the respective single set relationship. (> QuineVsRussell: several empty sets, there depending on type).
Solution/Daniel Nolan: (2001, Kaß 7, 2004): VsLewis, VsZermelo: empty set as real part of units:
Def "Esingleton" by A/Nolan: {A} consists of 0 and a thing {A} - 0 . (Terminology: "Singleton": only card of one color).
Esingleton/Nolan: similar assumptions apply to them as in Lewis' single sets.
Mixed Sum/Nolan: this problem becomes that of sums of 0 and atoms other than Esingletons. In Nolan, these are never elements of sets.
Object/Nolan: (2004.§4): only certain "big" things can be considered as 0. So all "small" things are allowed as elements of classes.
Individual/Nolan: many "little" things are individuals according to him among all Esingleton relationships.
Empty Set/Schwarz: all these approaches are not flawless. The treatment of the empty set is always somewhat artificial.
Schwarz I 90
Empty Set/Lewis/Schwarz: set of all individuals (see above): There is a good reason for this! ((s) So there are no individuals and the empty set is needed to express that.). Subset/Lewis/Schwarz: is then defined as disjunctive: once for classes and once for the empty set.
Possibilistic structuralism/Schwarz: is elegant. Vs: it prevents set-theoretical constructions of possible worlds (e.g. as sentence sets).
If you reduce truths about sets to those about Possibilia, you can no longer reduce Possibilia to sets.


1. David Lewis [1991]: Parts of Classes. Oxford: Blackwell
2. David Lewis [1993d]: “Mathematics is Megethology”. Philosophia Mathematica, 3: 3–23. In [Lewis
1998a]

Lewis I
David K. Lewis
Die Identität von Körper und Geist Frankfurt 1989

LewisCl I
Clarence Irving Lewis
Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991

Schw I
W. Schwarz
David Lewis Bielefeld 2005
Putnam, H. Field Vs Putnam, H. III 113
Pure Mathematics/Putnam: should be interpreted in a way that it asserts the possible existence of physical structures that satisfy the mathematical axioms. FieldVsPutnam: pure mathematics should not be interpreted at all.
I 211
Properties/Relations/Putnam: (1970): are predicative, according to them we have a few basic physical prop and rel from which all others are derived: 1st order: Allows no reference to a totality of physical objects when a new property is constructed.
2nd order: Allows reference to the totality of the properties of the 1st order.
3rd order: Allows reference to the totality of the properties of the 1st and 2nd order. - Every physical property appears on any level of the hierarchy -> functionalism.
Functional properties are 2nd or higher order properties - the prop that the role has may differ from person to person.
I 214
FieldVsPutnam: instead of properties provide instantiations of properties with steps.
I 268
Mathematics/Ontology/Putnam: ("Mathematics without foundations", 1976b, 1975 "What is mathematical truth?"): Field: Putnam Thesis: the mathematical realist does not have to accept the "mathematical object picture". He can formulate his views in purely modal terms. And that not as an alternative, but only as another formulation of the same view.
I 269
Indispensability Argument/Putnam: appear in the subsequent text. Field: If "Mathematics as a modal" logic was really an equivalent description of mathematics in terms of mathematical objects (MO), then it should also be possible to reformulate the Indispensability Argument so that there is a prima facie argument for one or the other kind of modalized mathematics and mathematical objects.
FieldVsPutnam: but Section 6 and 7 show that we cannot formulate the indispensability argument like that: it requires MO and modalized mathematics does not bring them forth.
VSVs: but beware: I have not studied all the possibilities.
I 269
FieldVsPutnam: his mathematical realism seems puzzling: Mathematics/Ontology/Putnam: Thesis: there is a modal translation of pure mathematics: he presents a translation procedure that turns mathematical statements into modal statements, one that transforms acceptable mathematical statements (E.g. axioms of set theory) into true modal assertions that include no quantification, unless it is modalized away. (I.e. ​​no mathematical entities (ME) in the modal statements).
I 270
FieldVsPutnam: two general questions: 1) what kind modality is involved here?
2) what benefit is the translation to have?
ad 1): Putnam thinks that the "object-image" (the starting position) and its modal translation are equivalent at a deeper level.
FieldVs: that’s really not interesting: "mathematically possible" should coincide with "logically possible" in any reasonable view (this is stated by conservatism). ((s) contrary to the above).
Important argument: if A is not mathematically possible, then "~A" is a consequence of mathematics - i.e. if A (and then also its negation) are purely non-mathematically, then "~A" is logically true.
If Putnam now says that his modal translation involves a "strong and clear mathematical sense of possibility", then a mathematical possibility operator must be applied to sentences that contain ME.
However, such a sentence A could also be a mixed sentence (see above, with purely mathematical and purely physical components).
I 271
FieldVsPutnam: for purely mathematical sentences mathematical possibility and truth coincide! But then the "modal translations" are just as ontologically committed as the mathematical assertions.
FieldVs"Mathematical Possibility"/FieldVsPutnam: we had better ignore it. Maybe it was about 2nd order logical possibility as opposed to 1st order for Putnam?
I 271
FieldVsPutnam: what benefits does his modal translation have? Does it provide a truth transfer (as opposed to the transmission of mere acceptability)? And what value has it to say that the mathematical statements are both true and acceptable? Etc. Mathematics/Realism/Putnam/Field: Putnam describes himself as
"mathematical realist": Difference to Field’s definition of realism: he does not consider ME as mind-independent and language-independent, but (1975):
Putnam: you can be a realist without being obliged to mathematical objects.
I 272
The question is the one that Kreisel formulated long ago: the question of the objectivity of mathematics and not the question the existence of mathematical objects. FieldVsPutnam: this is puzzling.
I 277
Model Theory/Intended Model/Putnam/Field: this morality can be strengthened: there is no reason to consider "∈" as fixed! Putnam says that in "Models and Reality": the only thing that could fix the "intended interpretation" would be the acceptance of sentences that contain "∈" through the person or the community. Putnam then extends this to non-mathematical predicates. ((s)> Löwenheim-Skolem).
FieldVsPutnam: this is misleading: it is based on the confusion of the view that the reference is determined, E.g. by causal reasoning with the view that it is defined by a description theory (description theory, (labeling theory?), in which descriptions (labels?) that contain the word "cause" should play a prominent role. (> Glymour, 1982, Devitt, 1983, Lewis 1984).

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Tarski, A. Field Vs Tarski, A. Brendel I 68
T-Def/FieldVsTarski: does not do justice to physicalistic intuitions. (Field 1972). Semantic concepts and especially the W concept should be traceable to physical or logical-mathematical concepts. Tarski/Brendel: advocates for a metalinguistic definition himself that is based only on logical terms, no axiomatic characterization of "truth". (Tarski, "The Establishment of Scientific Semantics").
Bre I 69
FieldVsTarski: E.g. designation: Def Designation/Field: Saying that the name N denotes an object a is the same thing as stipulating that either a is France and N is "France" or a is Germany and N is "Germany"... etc.
Problem: here only an extensional equivalence is given, no explanation of what designation (or satisfiability) is.
Bre I 70
Explanation/FieldVsTarski/Field: should indicate because of which properties a name refers to a subject. Therefore, Tarski’s theory of truth is not physicalistic. T-Def/FieldVsTarski/Field/Brendel: does not do justice to physicalistic intuitions - extensional equivalence is no explanation of what designation or satisfiability is.
Field I 33
Implication/Field: is also in simpler contexts sensibly a primitive basic concept: E.g. Someone asserts the two sentences.
a) "Snow is white" does not imply logically "grass is green".
b) There are no mathematical entities such as quantities.
That does not look as contradictory as
Fie I 34
John is a bachelor/John is married FieldVsTarski: according to him, a) and b) together would be a contradiction, because he defines implication with quantities. Tarski does not give the normal meaning of those terms.
VsField: you could say, however, that the Tarskian concepts give similar access as the definition of "light is electromagnetic radiation".
FieldVsVs: but for implication we do not need such a theoretical approach. This is because it is a logical concept like negation and conjunction.
Field II 141
T-Theory/Tarski: Thesis: we do not get an adequate probability theory if we just take all instances of the schema as axioms. This does not give us the generalizations that we need, for example, so that the modus ponens receives the truth. FieldVsTarski: see above Section 3. 1. Here I showed a solution, but should have explained more.
Feferman/Field: Solution: (Feferman 1991) incorporates schema letters together with a rule for substitution. Then the domain expands automatically as the language expands.
Feferman: needs this for number theory and set theory.
Problem: expanding it to the T-theory, because here we need scheme letters inside and outside of quotation marks.
Field: my solution was to introduce an additional rule that allows to go from a scheme with all the letters in quotation marks to a generalization for all sentences.
Problem: we also need that for the syntax,... here, an interlinking functor is introduced in (TF) and (TFG). (see above).
II 142
TarskiVsField: his variant, however, is purely axiomatic. FieldVsTarski/FefermanVsTarski: Approach with scheme letters instead of pure axioms: Advantages:
1) We have the same advantage as Feferman for the schematic number theory and the schematic set theory: expansions of the language are automatically considered.
2) the use of ""p" is true iff. p" (now as a scheme formula as part of the language rather than as an axiom) seems to grasp the concept of truth better.
3) (most important) is not dependent on a compositional approach to the functioning of the other parts of language. While this is important, it is also not ignored by my approach.
FieldVsTarski: an axiomatic theory is hard to come by for belief sentences.
Putnam I 91
Correspondence Theory/FieldVsTarski: Tarski’s theory is not suited for the reconstruction of the correspondence theory, because fulfillment (of simple predicates of language) is explained through a list. This list has the form
"Electron" refers to electrons
"DNS" refers to DNS
"Gene" refers to genes. etc.
this is similar to
(w) "Snow is white" is true iff....
(s)> meaning postulates)
Putnam: this similarity is no coincidence, because:
Def "True"/Tarski/Putnam: "true" is the zero digit case of fulfillment (i.e. a formula is true if it has no free variables and the zero sequence fulfills it).
Def Zero Sequence: converges to 0: E.g. 1; 1/4; 1/9; 1/16: ...
Criterion W/Putnam: can be generalized to the criterion F as follows: (F for fulfillment):
Def Criterion F/Putnam:
(F) an adequate definition of fulfilled in S must generate all instances of the following scheme as theorems: "P(x1...xn) is fulfilled by the sequence y1...yn and only if P(y1...yn).
Then we reformulate:
"Electron (x)" is fulfilled by y1 iff. y1 is an electron.
PutnamVsField: it would have been formulated like this in Tarskian from the start. But that shows that the list Field complained about is determined in its structure by criterion F.
This as well as the criterion W are now determined by the formal properties we desired of the concepts of truth and reference, so we would even preserve the criterion F if we interpreted the connectives intuitionistically or quasi intuitionistically.
Field’s objection fails. It is right for the realist to define "true" à la Tarski.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994

Bre I
E. Brendel
Wahrheit und Wissen Paderborn 1999

Putnam I
Hilary Putnam
Von einem Realistischen Standpunkt
In
Von einem realistischen Standpunkt, Vincent C. Müller Frankfurt 1993

Putnam I (a)
Hilary Putnam
Explanation and Reference, In: Glenn Pearce & Patrick Maynard (eds.), Conceptual Change. D. Reidel. pp. 196--214 (1973)
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (b)
Hilary Putnam
Language and Reality, in: Mind, Language and Reality: Philosophical Papers, Volume 2. Cambridge University Press. pp. 272-90 (1995
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (c)
Hilary Putnam
What is Realism? in: Proceedings of the Aristotelian Society 76 (1975):pp. 177 - 194.
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (d)
Hilary Putnam
Models and Reality, Journal of Symbolic Logic 45 (3), 1980:pp. 464-482.
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (e)
Hilary Putnam
Reference and Truth
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (f)
Hilary Putnam
How to Be an Internal Realist and a Transcendental Idealist (at the Same Time) in: R. Haller/W. Grassl (eds): Sprache, Logik und Philosophie, Akten des 4. Internationalen Wittgenstein-Symposiums, 1979
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (g)
Hilary Putnam
Why there isn’t a ready-made world, Synthese 51 (2):205--228 (1982)
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (h)
Hilary Putnam
Pourqui les Philosophes? in: A: Jacob (ed.) L’Encyclopédie PHilosophieque Universelle, Paris 1986
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (i)
Hilary Putnam
Realism with a Human Face, Cambridge/MA 1990
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam I (k)
Hilary Putnam
"Irrealism and Deconstruction", 6. Giford Lecture, St. Andrews 1990, in: H. Putnam, Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992, pp. 108-133
In
Von einem realistischen Standpunkt, Vincent C. Müller Reinbek 1993

Putnam II
Hilary Putnam
Representation and Reality, Cambridge/MA 1988
German Edition:
Repräsentation und Realität Frankfurt 1999

Putnam III
Hilary Putnam
Renewing Philosophy (The Gifford Lectures), Cambridge/MA 1992
German Edition:
Für eine Erneuerung der Philosophie Stuttgart 1997

Putnam IV
Hilary Putnam
"Minds and Machines", in: Sidney Hook (ed.) Dimensions of Mind, New York 1960, pp. 138-164
In
Künstliche Intelligenz, Walther Ch. Zimmerli/Stefan Wolf Stuttgart 1994

Putnam V
Hilary Putnam
Reason, Truth and History, Cambridge/MA 1981
German Edition:
Vernunft, Wahrheit und Geschichte Frankfurt 1990

Putnam VI
Hilary Putnam
"Realism and Reason", Proceedings of the American Philosophical Association (1976) pp. 483-98
In
Truth and Meaning, Paul Horwich Aldershot 1994

Putnam VII
Hilary Putnam
"A Defense of Internal Realism" in: James Conant (ed.)Realism with a Human Face, Cambridge/MA 1990 pp. 30-43
In
Theories of Truth, Paul Horwich Aldershot 1994

SocPut I
Robert D. Putnam
Bowling Alone: The Collapse and Revival of American Community New York 2000
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 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).

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994

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 Field I 18
- Mathematics / ontology / explanation / Field: Assume no mathematical entities because they are causally irrelevant.
I 24
yet: FieldVs causal theory (of knowledge)

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
Substantivalism Pro Field I 184
Substantivalism: per: Newton (bucket experiment) -
I 191
Substantivalism / Field: per: because it does not need a mathematical entities - VsRelationism: needs moderate Platonism - therefore uninteresting.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994

The author or concept searched is found in the following 5 theses of the more related field of specialization.
Disputed term/author/ism Author
Entry
Reference
Platonism Field, Hartry I 44
To succeed in VsPlatonism, we must also show, thesis: that mathematics is dispensable in science and metalogic. Then we have reason not to literally have to believe in mathematics. (>Indispensability argument).
I 45
If that succeeds, we can get behind the agnosticism.
I 186
Def moderate platonism/mP/Field: the thesis that there are abstract objects like numbers. Then one probably also believes that there are relations of physical size between objects and numbers. (But only derived): Example "mass in kilogram" is then relation between a given physical object and the real number 15,2.
Example "distance in meters" is a relation between two objects ((s) on one side) and the real number 7,4.
The difference to high-performance platonism (HPP) lies in the attitude to these relations:
Moderate Platonism: Thesis: These are conventional relations derived from more fundamental relations existing between physical objects alone.
Def High Performance Platonism/Field: denies that and takes the relations between objects and numbers as a bare fact that cannot be explained in other terms.
Inflated one could explain this as "platonistic participation".
II 332
Standard Platonism: Thesis: Mathematical theories such as set theory or the theory of real numbers are about different mathematical domains, or at least about certain structures, because there is no need to assume that isomorphic domains (i.e. domains with the same structure) would be mathematically indistinguishable. Thus, "regions" should not be assumed as sets.
II 333
Def "Platonism of perfection": (plenitude): postulates a set of mathematical objects. Thesis: Whenever we have a consistent purely mathematical theory, there are mathematical objects that fulfill the theory under a standard-fulfillment relation. Platonism of perfection: but also suggests that we can consider all quantifiers about mathematical entities in this way,
I 334
that they are implicitly limited by a predicate to which all other predicates of mathematical entities are subordinated: The "overarching" predicate: is then different between the different mathematical theories. These theories then no longer conflict.
II 335
Universe/Standard Platonism/Field: (Thesis: "Only one universe exists"). Problem/PutnamVsPlatonism: how do we even manage to pick out the "full" (comprehensive) universe and confront it with a sub-universe, and accordingly the standard element relationship as opposed to a non-standard element relationship? (Putnam 1980). (Here placed from the perspective of "Universe").
Putnam: Thesis: We simply cannot do that.
Physics Field, Hartry I 6
Thesis: it must be possible to practice theoretical physics without appealing to mathematical entities.
I 71
Physics/Field: Thesis: All you need are normal physical objects and an entity with geometric and electromagnetic properties.
III XI
Thesis: the conclusions that can be made with the assumption of theoretical physical entities cannot be made without them. They are theoretically indispensable. No physical theory is possible without theoretical entities.
Modalisation Putnam, H. Field I 268
Field: Putnam Thesis: The mathematical realist does not have to accept the "mathematical object picture". He can formulate his view in purely modal terms. And not as an alternative, but only as a different formulation of the same view.
I 296
Mathematics/Ontology/Putnam: Thesis: There is a modal translation of pure mathematics: he presents a translation procedure that transforms mathematical statements into modal statements, one that transforms acceptable mathematical statements (e.g. axioms of set theory) into true modal statements that contain no quantification unless it is modalized away. (So no mathematical entities in the modal statements).
I 270
FieldVsPutnam: two general questions: 1. what kind of modality is involved here?
2. what is the benefit of the translation? ...+...
I 275
Modal Translation/Field: Thesis: Modal translation of single mathematical applications without assumption of mathematical entities is easier than one of the whole mathematics, because the applications do not require pure mathematics.
II 321
Mathematics/Modal/Modality/Putnam/Field: (Putnam 1967, Hellman 1989): Thesis: Mathematics should be understood modally. ((s) "There is one possible world where the power of the continuum is so and so great, and another where it is greater/smaller?). Field: even if there are no mathematical entities at all, it could not be the case that for a value of a for the Ca ("The thickness of the continuum is Aleph a") modally interpreted is objectively true.
FieldVs: . +
Horwich I 398
Set/Putnam: (elsewhere) thesis: speech about sets can always be translated into speech about possibilities.

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994

Horwich I
P. Horwich (Ed.)
Theories of Truth Aldershot 1994
Mathematics Putnam, H. Field II 319
Putnam: Thesis: there are many properties and relations in which these mathematical entities can stand to each other. And there is not much to say about what such properties and relations for which we use our mathematical predicates should stand, apart from making the mathematical propositions we accept true.
II 321
Truth/Mathematics/Putnam: Thesis: Truth is too easy to attain ((s) by reinterpretation) to limit our choice of axioms. (However, only as long as there are (infinitely many) mathematical objects).
II 328
Usefulness/truth/mathematics/Putnam/Field: (Putnam 1971 locus classicus, unlike 1980): Thesis: We must regard mathematics as true in order to be able to explain its usefulness in other fields. E.g. in science and meta logic. (i.e. the theory of the logical sequence). Modality/modal/mathematics/Field: this contrasts with his earlier view that we can use modality instead of mathematical objects to explain mathematical truth.
II 329
Modal explanation: will not work for other disciplines like physics. (FieldVsPutnam, Field 1989/91: 252-69). Putnam/Field: the general form of its argument goes like this:
(i) we must speak in terms of mathematical entities in order to practice science, meta logic, etc..
(ii) if we need them for such important purposes, we have reason to believe that this kind of entities exists.
VsPutnam/Field: ... +

Field IV
Hartry Field
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994
mathemat. Entities Wright, Cr. Field I 23
Wright (s): our standard use is not sufficient for the determination of mathematical entities. (FieldVsWright).
  I 24
  VsWright: but that this could be true also 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. 553-67
In
Theories of Truth, Paul Horwich Aldershot 1994