“What is Applied Mathematics?”  James Brown

Three questions:

1.      How does mathematics ‘hook onto’ the world?

2.     Are some objects referred to in theories ‘merely mathematical’, or do they exist in their own right?

3.     Is math essential for science?

Re 1:  Measurement theory says that mathematical representation of a non-mathematical realm occurs when there is a homomorphism between a relational system P and a math system M – a structure preserving mapping from the domain, D, of the relational system P to the domain D* of the mathematical system M.  Mathematics hooks onto the world by providing representations in the form of structurally similar models.

Representing objects vs representing properties of objects; the latter allows for physically uninstantiated values, but commits us to platonic properties.

Re 2:  There are mathematical ‘mere artifacts’ – e.g., the average family.  But compare Maxwell’s electrodynamic fields; quantum states; and space-time manifolds.  The issue of the independent existence of mathematically modeled objects must be evaluated on a case-by-case basis.

Re.3:  Quine/Putnam vs Field  pp. 27-28

Field’s conservatism argument for the dispensability of mathematics.  If a conservative nominalization of the mathematical content of an empirical theory exists, then that shows that the math is dispensable, because a ‘conservative’ nominalization is one that has all of the predictive consequences of the original theory.

Objections to this argument:

1.      The notion of logical consequence needed is that of second-order logic, which isn’t recursively axiomatizable , so the notion of consequence must be semantical and therefore set-theoretic.

2.     The need for mathematical models not just for talking about how things are, but to talk about what is and isn’t possible, given certain conditions.

3.     Math is also needed methodologically, for the creation and comprehension of physical theories.

New question:  Could mathematics actually describe the world directly rather than merely represent it?

 

The Quine/Putnam vs Field debate can be cast as a debate about this.

p. 30 bottom: The autonomy of mathematics supports the representationalist account.

 

The debate about structuralism can also be related to this new question.  According to structuralism, math can apply directly to the world, by contrast with platonism, according to which math is transcendental and merely representational.

 

But, asks Brown, are structures discovered in the world, or to recognize X as having structure S must we already have the idea of the structure S itself?

 

Against structuralism:

 

Zermelo and von Neuman structures for the natural numbers are different structures, because all properties are essential in mathematics.

 

Every consistent mathematical structure has an instantiation, but not every such structure has a physical instantiation.  (Says who?: Plato!)

 

The foundational notion of set (comp. e.g., to the notion of groups) is contrary to the spirit of structuralism, because the identity of a set is defined in terms of its members.

 

(Last para: spoken like a true ‘pi in the sky’ space cadet!  J)

 

 

Mark Balaguer: “A Fictionalist Account of the Indispensable Applications of Mathematics”

 

The applicability of math is a problem for Platonism, because mere truth of Platonism would not be enough to explain its applicability;  relevance is also required.  Why is mathematical theory, platonistically construed, relevant to physical theory? 

PCI “The Principle of Causal isolation”:  There are no causal interactions between mathematical and physical objects.”

Balaguer:  PCI must be true, because we cannot make coherent sense of causal interactions with sets, as distinct from their (say) concrete members, contra Maddy (1990).

PCI makes it hard for platonists to explain the relevance.  PCI also makes it hard for fictionalists who accept PCI to explain the relevance.  There are some anti-platonists who reject PCI; however, this turns out in general to be problematic.

One idea platonists might try is to posit mathematico-physical facts: e.g., the fact that that a system S is currently at 40 degrees Celsius.  But PCI implies that this mixed fact supervenes on a bottom level purely physical fact about S and a purely mathematical fact about 40. (see argument in n. 10)

This could lead the Platonist to adopt a representational, rather than causal, explanation of the applicability .  On such a view, e.g., given that physical temperature states are related to each other analogously to the way real numbers are related to each other, it is convenient to represent the bottom-level physical state using the real numbers, rather than trying to spell out the physical details of its temperature state. 

 

 

But there is a worry that this representational account may not be able to handle all questions of mathematical applicability.  While all applications that that can be explained by the representational account are thereby dispensable (argument given on p. 295), those that aren’t are so far still indispensable, and therefore in need of a different account.

Proposal: 

(APP)  All mathematics ever does in empirical science is provide theoretical apparatuses (or in other words, conceptual frameworks) in which to make assertions about the physical world; i.e. math is not relevant to the operation of the physical world, only to our understanding  of it. 

This is akin to the representational account except for not requiring that a nominalized version of the physical theory – one from which references to mathematical entities  has been removed via some appropriate representation theorem  – be available.  Still, what would be articulated about the world from the standpoint of such a theory is precisely what the theory is about, even if we don’t yet know how to articulate it.  Although we can sort of articulate bits of it piecemeal.   So we should not doubt that, say, Quantum Mechanics has a nominalistic content which captures its complete picture of the world, even if we do not know yet how to formulate it.

Note:  Fictionalism can appeal to APP as well in its explanation of mathematical applicability.  E.g., QM per se never entails that Hilbert spaces are causally relevant to the physical world.  And certainly the use of Hilbert spaces as an heuristic device does not require that reference is being made to platonic objects in the use of Hilbert spaces.

 

ULRICH MEYER:  “HOW TO APPLY MATHEMATICS”

1.  Intro.

For physics to genuinely apply mathematics, but not conversely, there must be a sharp difference in the role of mathematical and physical objects in physical theories.  Neither nominalists nor Quine/Putnam indispensability realists are able to say what that is.  But a less expressively restrictive indispensability realism can.

2. The Indispensability Argument.

The argument aims to refute the nominalist by showing that mathematical objects are indispensably appealed to in physical theory, and in a way that marks no significant difference between the theoretical contributions of physical and mathematical objects.  Its background assumptions include (A1): the regimentation of physical theory to extensional 1st order languages; and (A2): the commitment to a mere finitude of physical objects.  But (A1) makes for an awkward treatment of quantification over masses and distances in Newton’s theory, requiring them to be treated as relations between physical objects and numbers.

e.g., Object a has mass n kilogram à Man

But the idea that mass is a relational property of objects seems forced, and the role of the numbers here seems to have nothing to do with mass per se, but just to serve as a kind of index.  Physical objects might be used as the indices instead of numbers, except for the fact that there are not enough of them.  So, in Newton’s theory, formulated in a 1st order way, we need mathematical objects as theoretical relata to serve as these indices.  “Man” is thus an indispensably “mixed” atomic formula, containing terms for both mathematical and physical objects, and there is no principled way to isolate its physical from its mathematical content.  Field tries to get around this by denying (A2) and Hellman and Chihara try to get around this by denying (A1), but for the wrong reasons.

 

 

3.  Mathematics and Infinity

Suppose that physics were finite.  Then we wouldn’t need math to formulate it.  But physics is infinite.  [But is it merely denumerably infinite or is it indenumerably infinite?]  If we had the cognitive resources of, say, God , then we wouldn’t need math either.  But we don’t.  The main purpose of math is to boost our expressive capacities in dealing with complex physical systems.  There are similarities (both serve to simply theory) and differences (math has to do with expressive capacity – ‘ideology boosting’) of role between mathematical objects and physical objects in physical theory, but such differences cannot be accounted for in a 1st order way.  At the same time, the fact that mathematical and physical objects enter physical theory for different reasons does not imply that they should be given different ontological status.

4.  Mass Properties

Intuitively, mass is an intrinsic property of physical objects, not a relation to something else (e.g., a number).  2nd order property quantifiers, modal operators, and a certain partial ordering among properties, can be used to give a characterization of that property. 

What we want is a characterization of the intrinsic mass property, X, had by a, such that a has the property X if and only if Man for some particular n.  There turn out to be some technical issues of distinguishing this intrinsic mass property in a 2nd order way from properties that are not equivalent to mass.  The property of having a certain amount of mass is different from the property having electron charge, for instance.  While all objects with electron charge have electron mass (i.e., the amount of mass had by an electron at rest), the converse does not follow.  And while having electron mass is having a mass property, having the conjunctive property of being red and having electron mass is not. 

So, it is not enough to say that there is some property X such that x has it and anything z is such that if z has X then Mzn.  Let n be the numerical value of electron mass, and suppose that only electrons happen to have that value of mass.  Then X so characterized would not distinguish between electron charge and electron mass.  One can avoid this by strengthening the above characterization of X to rule out such coincidences, by requiring for all z that if z had X then Mzn would be the case.  This replaces the material conditional with a counterfactual conditional.  But the result still doesn’t rule out the conjunctive property of being red and having electron mass: anything that had such a conjunctive property would have a mass property; but the conjunctive property is not a mass property per se.

We have now arrived at the most technical moment of the paper.  We are to suppose that properties may be partially ordered in the following way: a property X will be deemed less than or equal to a property Y just in case, necessarily, any object z that has X will also have Y.  Now Let ‘Ø [X]’ be an open sentence in property variable X.  It can be an atomic sentence, or a conjunctive open sentence, or whatever.  Then we can say, where X is mass, that the maximal property of mass, call it ‘MX [ØX]’ will be the property Y such that any property Z that satisfies the open sentence is contained within Y.  But the conjunctive property of being red and having electron mass does not satisfy the open sentence, and so is not contained  within it.

So, then, to say that a has the property of n kilograms of mass would be to say that a has the maximal monadic property MX such that:  [were-any-z-to-have-X,-then-Mzn-would-be-the-case]x.  This, Meyer claims, allows us to represent mass properties as the intrinsic properties we take them to be.

One thing to note about this 2nd order characterization is that although n is used in the characterization of the maximal property, it is not in a grammatical position such that it is available for objectual quantification.  I have tried to make this obvious with the use of the dashes and square brackets.  But we will see shortly, that Holland is not really wanting to make an ontologically deflationary point here.  [Though others might want to.]

Another thing that Meyer claims at this point is that when we do express a relation between a physical object and a number, as in the 1st order ‘Man’, the relation should be regarded as an internal relation, not an external relation, and that the 2nd order rendering helps us to see this.  To say that the relation is internal is to say that it’s obtaining or not is fully determined by the inherent properties of the relata. Spatial distance between things, as treated in, say, Newtonian mechanics is an external relation.  The quantity of mass of an object, expressed as a relation to a number, is an internal relation.  On p. 23 Meyer makes the claim that the relations that physical objects bear to mathematical objects are always internal.  What this amounts to is the claim that for every “mixed” 1st order relational statement of a physical theory, we can formulate it in 2nd order theory as the claim about a second order n-adic property had by some physical objects, where the references to mathematical objects now appear in the characterization of the property and are therefore not available for quantification.  He calls these structures “separation postulates”, because they show an important difference of function between the physical objects and the mathematical ones.  The theory is straightforwardly committed to the existence of the physical objects: reference to them occurs in existentially quantifiable positions.  The theory is not thereby committed to the existence of the mathematical objects, at least when referred to in a seemingly referentially opaque position as part of the characterization of an intrinsic property.

5.  Physical Content

And this observation is also key to how Meyer wants to effect a separation between the empirical content and mathematical content of a physical theory, without this being nominalistic or in aid of nominalism: just formulate the salient “separation postulates”, which show the mathematics as contributing to the ‘form’ of the theory, not its objectual content. 

Quine was right, he says : A theory T is ontologically committed to an object x just in case the theory logical entails x’s existence.  And a theory is physical just in case it logically entails the existence of some physical objects.

But now, says Meyer, let us say that:

A physical theory T applies an (e.g., mathematical) object x just in case (i) T is ontologically committed to x, and (ii) there is a theory T’ that is not committed to x’s existence, but which has the same physical content as T.

Note: the above does not imply that the theory could be formulated without reference to all mathematical objects, just without reference to x.  [Hmmmm.  Why formulate it in that way??  Maybe because he wants to formulate it in a way that causes the least problem for a platonist, while still providing a characterization of distinctness of function for math objects.] Meyer then claims (bottom p. 25) that a theory applying an object in this sense would show that the object’s only function is to help express its physical content – its only role is ‘ideology-boosting’.  [Note: this compares quite well with the account of empirical math applications proposed by Balaguer.]  But then, on the top of p. 26, he says: but wait, there’s more!  This only works if there are indeed mathematical objects after all. You cannot apply an object if there is no object to apply! “The numbers and functions referred to by the separation postulates need to exist for the expressions within the brackets to pick out the relations that we want them to pick out.   The theory’s physical content is only part of its deductive closure.  The rest of the theory, which makes ample reference to mathematical objects, is not superfluous.  It plays an essential role in expressing its physical content.  Hence this account of applied mathematics does nothing to advance the case of the mathematical nominalist.”  [But how do we get at “the rest of the theory”? By reverting to its 1st order formulation?  Or by treating all of the erstwhile opaque 2nd order contexts in the definitions of physical properties and separation postulates as transparent after all?  It cannot be the latter, because then the explanation offered of the role of the math collapses.  So the proposal must be to keep both formulations of the theory:  one, T’, for account of the special role of math it provides, the other, T,  for the ontological commitment to math entities it provides.  But that is a little odd.  What, for instance, if the math applied is mathematical properties, rather than mathematical objects? Then we will seemingly need the 2nd order version of the theory for both tasks.  Also, and independently, why couldn’t fictional mathematical objects or properties serve here, a la Balaguer?  Apparent answer: because we want claims about mathematical objects to be true.  But maybe when Meyer says that his scheme does nothing to advance the case of nominalism, he is putting it too strongly.  Perhaps he should have said: this account of applied mathematics does not force us to be nominalists – although it does not force us not to be either.]

“HOW APPLIED MATHEMATICS BECAME PURE”, Penelope Maddy

--the history of applied and of pure mathematics, the changing relations between them, and philosophical lessons to be learned --

Sometimes an historical perspective can be quite revealing.  Sometimes it should give us philosophical pause.

A historical reversal, in broad strokes, of philosophical regard for mathematics:

1. from Plato, who thought that our rational quantitative inquiries yield  true knowledge of eternal, transcendental  forms, while our empirical qualitative inquiries yield at best defeasible opinion about the immanent passing world;

2. to the scientific revolution of the 15th and 16th centuries, in which mathematics ‘became one’ with science; became ‘the language in which the book of nature was written’.  The subject matter of mathematics became immanent, in Aristotle’s (and maybe Pythagoras’s) sense.  Empiricism was ascendant over rationalism.  (Yadayada…)  Scientifically good descriptions of the phenomenon being studied should be quantitative, or mathematical; and nature, through observation and measurement, provides us with the data and means to formulate them.  Nature guides us to the correct mathematics, and to its correct worldly applications.  Science and mathematics develop together in a close symbiotic relation;

3. to the present day: where scientific knowledge is the best, is the standard of knowledge; epistemology is naturalized, a la Quine.  Yet though the science is ever more highly mathematized, the status of the math, both in this role and independently of this role, has become philosophically problematic.

What happened between 2 and 3?

·        The 19th century saw mathematicians begin to introduce concepts that did not -- by their lights at the time -- correspond to anything in the empirical world.  They had no physical interpretation.  These concepts were introduced and explored for purely mathematical reasons.  The methods used were those also divorced from empirical methods.  Mathematics as a discipline began to assert its autonomy -- in terms of what it studied and how it studied it-- from empirical science.  Mathematics exploded into a hundred branches – “maths”.

 

·        The discovery by mathematicians of non-Euclidian geometries in the 19th century (discovered while exploring an uneasiness – theirs and Euclid’s -- about the Parallel Postulate), and then later the empirical discovery that physical space was non-Euclidian, led to the mathematical study of various geometries of various abstract spaces, to recognition that the geometry of physical space was an empirical question, but at the same time to the protection of Euclidian geometry by mathematicians as a true geometry, not about physical space but about an abstract space. Michael Resnik, a philosopher, refers approvingly to this positing of an abstract realm as the subject matter for Euclidean geometry as a “Euclidian rescue”. Generalizing this sort of move gave mathematics its own protected subject matter.  Science was of course still free to visit (if it asked politely) the increasingly well-stocked warehouse of mathematical theories, to try to find the right mathematical tools for representing or modeling aspects of the empirical world.

 

·        The acceptance of atomic theory in the 20th century created more demands on mathematics.  There was by now no longer any issue of the value and need for mathematics for scientific purposes.  But there was a shift in mathematical modeling of empirical phenomena under study, from descriptive accuracy, to approximation and idealization, sometimes quite pronounced idealization, sometimes idealization without any very clear independent understanding of the nature of what one was trying to represent, and so without any clear understanding of why the approximations and idealizations worked when they worked: e.g., comp. the theory of heat, quantum mechanics, particle physics, the structure of space-time, black holes,  etc..  So math in application in these domains was not necessarily giving us, or even trying to give us, an explanatorily accurate picture of the empirical phenomena being modeled.  There was no going back to pre-revolutionary qualitative science, of course.  But must we humor the mathematician, and allow them to go back to Plato, to regard their subject matter as literal platonic abstractions?  What explanatory or modeling purpose is served by that?  Or is it only anachronistic philosophers who want to do that?

WHERE WE ARE NOW, ACCORDING TO MADDY

·        WE ARE NOT TRYING TO DISCOVER MATHEMATICS IN NATURE LIKE GALILEO AND EULER AFFECTED TO DO.             WE ARE ‘CONSTRUCTING’ IT.  BUT WHAT MOTIVATES US TO CONSTRUCT WHAT?  THE CONTRAINTS AND THE MOTIVATIONS SEEM TO BE LARGELY INTERNAL TO THE DEVELOPMENT OF MATHS BY MATHEMATICIANS.

 

·        THE REASONS FOR DISCREPENCIES BETWEEN MODEL AND MODELLED, TO THE EXTENT THAT WE UNDERSTAND THEM, ARE SO DIVERSE THAT THE PROSPECT OF A UNIVERSAL ACCOUNT FOR APPLICABILITY OF MATHEMATICS, AS OPPOSED TO A CASE-BY-CASE APPROACH, LOOKS DOUBTFUL.

 

·        WITH THE COMPLETE DIVORCE OF THE METHODS OF PURE MATHEMATICS FROM THE EMPIRICAL METHODS OF SCIENCE, IT WOULD SEEM THAT THE RATIONAL AUTHORITY, THE OBJECTIVITIY, OF THE FINDINGS OF PURE MATHEMATICS MUST SOMEHOW RESIDE IN ITS DISTINCTIVE METHODS, AND SO TO UNDERSTAND WHERE THAT AUTHORITY COMES FROM AND WHAT IT COMES TO WE NEED TO UNDERSTAND THOSE METHODS.  HOW DO WE GO ABOUT THEIR STUDY?

 

SET THEORY AS A BROAD FOUNDATIONAL FRAMEWORK, EXEMPLIFYING MATHEMATICAL METHODS AND THE PROBLEMS OF UNDERSTANDING THEM:  AXIOMATIZATION; CRITERIA OF COHERENCE; STANDARDS OF DEFINITION, PROOF; UNANSWERED QUESTIONS.  WHAT GUIDES ITS FUTURE DEVELOPMENT?  PERHAPS SIMPLY THE WELL-INFORMED INTUITIONS OF SET-THEORETIC PRACTITIONERS, CONSTRAINED BY COHERENCE, AND ESPECIALLY MATHEMATICAL FRUITFULNESS?  IS THIS A KIND OF ‘MATHEMATICAL OBJECTIVITY WITHOUT MATHEMATICAL OBJECTS’?  IF SO, MADDY HAS ARRIVED AT IT, CONTRA FIELD, LARGELY THROUGH HISTORICAL REFLECTION, NOT THROUGH HIGHLY ABSTRACT  PHILOSOPHICAL ARGUMENTS AGAINST THE POSSIBILITY OF RATIONAL BELIEF IN SOME FORM OF MATHEMATICAL REALISM

 

“MATHEMATICAL OBJECTIVITY AND MATHEMATICAL OBJECTS”

Hartry Field (1998)

Initial Question:  Must (putative) objectivity of mathematical claims be explained in terms of the existence of mathematical objects, or can it be explained in some other way…e.g., via an interpretation of mathematical claims as concerning possibilities?  (p. 388)

Thesis to be defended: The deepest issue with a variety of versions of platonism including standard ‘objectual’ platonism, is not about the existence of abstract objects at all, but about mathematical objectivity and its limits. And that objectivity can be grounded in logic: 1st order logic, supplemented by a formal account of finiteness (p. 401)

1.      Logical Objectivity and Specifically Mathematical Objectivity

Will assume that logic, and hence mathematical proof, is fully objective (qua validity).  If that is all there is to mathematical objectivity, then there is no requirement of mathematical objects (because logic is formal in the sense that l-validity and l-truth do not per se imply the existence of any objects.

But what about the purely mathematical aspects of math imbodied in its axioms, the ultimate starting points of all mathematical proofs.  What is objective about these?

Suppose that their objectivity is just a matter of their being accepted by us, or by mathematicians.  It turns out that that leaves open the possibility (and actuality) that there will be mathematical claims such that neither they nor their denials are provable from logic plus the accepted mathematical axioms.  And thus neither they nor their denials will be objectivity correct.  In fact either they or their denials will be consistent with our best mathematical theory, closed under logic.

An important current example is a claim about the size of the continuum: it will be indeterminate whether its size is aleph1, aleph2, aleph3….and into the transfinite. The mathematical axioms that are currently accepted do not imply that there is an objectively correct size of the continuum.

[Field wants to contrast this example with Godel’s proof that there are undecidable sentences of the form

            (*)  For all natural numbers x, Bx; in which ‘Bx’ is a decidable predicate

Why?]

2.      Mathematical Objects Without Specifically Mathematical Objectivity.

Following Hilary Putnam, even if there is a fixed universe of mathematical objects, there need be no objectively correct answer to the question of the continuum.  This is because whatever our choice of answer (aleph1, aleph23), we will be able to find properties and relations with which to interpret our theory so that on our choice it comes out true (Why? Cf. p. 391).  So this suggests that there are limits to the objectivity of mathematics as it now stands.

[Field alludes to a determinate but primitive notion of finiteness, which, when added to standard 1st order logic, would block an analogous result for Godel’s undecidable sentences of number theory, so that at least number theory per se would remain completely objective.]

Field then sketches (pp. 391-2) what he calls a ‘moderately non-objectivist’ picture of mathematical methods, which turns not on truth but on consistency, plus interest, beauty, utility relative to context, etc..

3.      The Prospects For Mathematical Objectivity Without Mathematical Objects.

Interpreting the claims of mathematics in some non-standard, non objectual way (e.g. as modal claims) will not help avoid the result in 2 above.  Nor can it be convincingly argued that logic even supplemented with the account of finiteness somehow depends on prior mathematics in a way which undercuts the result.

 

4.     The Existence and Nature of Mathematical Objects.

Let ‘platonism’ imply the view that there is a unique correct answer to the continuum problem. (‘unqualified objectivism’)

            Forms of anti-platonism:

                        Conceptualism

                        Constructivism

                        Balaguer’s ‘Full-blooded Platonism’ (Why?).

                        Fictionalism (no math objects)

“…it might…be argued that the difference between the anti-platonist views and the standard Platonist view of a single universe of mind-independent mathematical objects collapses, if the standard Platonist accepts Putnam’s argument and recognizes the futility of thinking that those mathematical objects will supply a kind of objectivity that is unavailable on the anti-platonist views” (p. 395)

5.      The ‘Access’ Argument. (Benacerraf (1973)) defended, with important exemptions noted on p. 397 top.

 

6.     A number of possible ways of elaborating ‘The Structuralist (Benacerraf (1965)) Insight’ are distinguished and critiqued.

 

7.     Mathematical Objects and the Utility of Mathematics

Must we speak of numbers in doing our best science? (Can a nominalization project succeed?)

Even if so, does this give us a reason to suppose that numbers exist? (Would that be only if numbers are taken to be ‘causally involved’ in producing physical effects?)

 

“INDISPENSIBILITY ARGUMENTS AND MATHEMATICAL REALISM”

DAVID GABER (UNPUB. MS., 2009)

            To be argued:  that the Quinian Indispensability Argument is in tension with the realities of mathematical and scientific practice.  The mere indispensability of an entity to some theoretical discourse is not grounds, per se, for taking such entities as having any kind of reality independent of such discourse.  (p.1)

            The QI Argument:

1.  We ought to have ontological commitment to all and only those entities that indispensable to our best scientific theories.

2.  Mathematical entities are indispensable to our best scientific theories.  Therefore

3.  We ought to have ontological commitment to mathematical entities.  (p. 2)

Problems understanding the explanatory role of abstracta in empirical contexts. (p.3)

Explanation and Theoretical Entities

Compare: “The acceleration due to gravity at sea level on Earth is 9.8 m/sxs” with “There exists a real number x such that the magnitude of the acceleration due to gravity at sea level on Earth is x m/sxs”

Note: the latter does not by itself imply that the said number exists independent of mind and language.

The chess analogy vs. ‘the mathematics game’?

pp. 5 – 6:   unexpected consequences of mathematical claims for empirical phenomena; and also for math itself via the interconnected nature of math, make mathematics seem rather un-game like.

But that doesn’t necessarily council platonism;  the prospect of fictionalism may still be pragmatically justified.

Mathematics in Scientific Explanations

A received view: the role of math as: the explicit rendering of properties of elements of scientific ontology and their causal interrelations.

Speaking of scientific explanations in general terms (e.g., the D-N model)  vs. speaking of the details of particular explanations.

p. 8  But it might be asked what the philosophical relevance is of those details.

In response:  Mark Wilson quote. 

A minor change in the physical nature of an explanation may have dramatic implications for an explanation.  The details can be critical.

Application of Math in Scientific Reasoning

p.9 The math used in modeling a system need not directly represent the activities and entities of the system in order to explain things about it – contra the received view.

            e.g., sliding block on an incline plane

p. 10  n.9:  ‘ensemble behaviors’ , in which the detailed underlying causal story is explanatorily irrelevant.  Other examples  . 10-11 

geometric (topological) relationships vs. causal relationships

geometric relationships can in many cases remain explanatory even if we change the underlying causal processes

pp. 12-13 Why is the mathematics involved in an explanation so stubbornly independent of the physical nature of the system?

But mightn’t that be taken as an indication of platonism?  No, and it would not account for the role of mind and language dependent facts in the choice of which math to deploy in which circumstances, in explaining the behavior of a given system.

Methodology of Mathematicians and Physicists.

Disparities in the notions of rigor employed.  Proof and argument vs. consistency with observation, explanatory utility, plausibility given known physical laws.

e.g., the Dirac delta function; the Feynman path integral.   Physicists prescribe their restricted deployment rather than outright rejection. And their instrumental deployment as such may be indispensable, notwithstanding their mathematical incoherence.

n.15  A compelling argument in favor of a unified notion of rigor would be strong evidence for mathematical Platonism. (Why, exactly?)

p. 15  Davey: inferential permissiveness vs inferential restrictedness.

p. 16 bottom:  a nuanced evaluation of the differences of rigor.

What Kind of Entities Are mathematical Entities?

p. 18  The role of experimental outcomes in underpinning our commitment to the existence of electrons?  What is the analog of that supposed to be in the case of mathematical entities appealed to in theoretical explanations?

pp. 18-19:  on the multiplicity of disparate maths used to describe the same system.

Maddy quote.

The “artificiality” of the role of mathematics in the sciences.

p. 20 top “The world does not appear to have a strong bias towards any particular mathematics.”

Revisiting the Indispensibility Argument:  Reject premise 1.

Background Considerations on Indispensibility Arguments:

They all seem to presuppose scientific realism as an interpretive stance towards the empirical content of scientific theories.  That raises the question of whether such arguments are biased towards mathematical realism as opposed to fictionalism.

Quine’s first premise is a bi-conditional.  Gaber doesn’t comment on the two directions of the bi-conditional considered separately, but the ‘only if’ clearly depends on Quine’s naturalism, and Quine supports the ‘if’ direction by appeal to his confirmation holism about theories, which has the mathematical aspects of the theory and their ontological consequences being accepted on the basis of the same empirical confirmation as the rest.

Quine’s second premise depends of course on having ruled out any nominalistic reduction of the mathematical aspects of a theory to something that leaves them out.

Michael Resnik offers a different indispensability argument for the truth of mathematical claims (see his “Scientific vs mathematical Realism: the Indispensibility Argument”, in Philosophia Mathematica Vol 3 (1995) pp.166-174).  He takes it as independent of any appeal to conformational holism.  It could also be argued that it is also independent of whole-sale Quinian naturalism.  Here it is:

1.  In stating its laws and conducting its derivations science assumes the existence of many mathematical objects and the truth of much mathematics

2.  These assumptions are indispensable to the pursuit of science; moreover, many of the important conclusions drawn from and within science could not be drawn without taking mathematical claims to be true.

3.  So we are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true.

4.  But we are justified in drawing conclusion from within science, since this is the only way we know of doing science, and we are justified in doing science.

5.  Therefore we are justified in taking the math used in science (including existence claims) to be true.

 

“FICTIONALISM, THEFT, AND THE STORY OF MATHEMATICS”,  Mark Balaguer

Wants to defend a certain kind of fictionalism against certain objections, but does not ultimately want to endorse either fictionalism or platonism.  His final view (not developed here) is a kind of ‘no-fact-of-the-matterism’ as regards platonism and fictionalism in mathematics.

‘Generic Fictionalism’:  Since mathematical theories purport to be about abstract objects, but there aren’t any such objects, mathematical theories are not literally true.

Fictionalism is a reaction to mathematical platonism.  Here is an argument for

‘Generic Platonism’:

1.      The sentences of our mathematical theories seem true.

2.     They should be taken at their grammatical face value.

3.     If we take them to be true at their grammatical face value, then we must be committed to the existence of the objects they purport to be about.

4.     But those would-be objects are abstract.

5.     So we must be committed to the existence of those abstract objects

2-4 can be defended empirically via semantic considerations.

But the conjunction of 1 and 2 does not give us the antecedent of 3.  The argument actually looks to be invalid (not what Balaguer had in mind).

There is the Quine/Putnam Indispensibility Argument for platonism.  But the indispensability can either be denied via nominalist reconstruals (thereby denying 2) or failing that, explained via fictionalism, e.g., an account of the role of mathematics in science as a descriptive or representational aid.

Re. disanalogies with fiction:  Better names for fictionalism might be ‘reference-failurism’, or ‘not-truism’.  Fictionalists do not need to make any substantive claim about similarities between math and fiction.

 

Objectivity and Correctness:

Truth vs truth in the story of mathematics. 

But the objectivity of math might be thought to outstrip math’s story, whether or not one ultimately takes the story for reality or fiction. It will be important to sort this issue out because, the main strategy for the defense of fictionalism adopted here is to mimic platonism:

Theft-Over-Honest-Toil Fictionalism (‘T-fictionalism’):  A sentence of pure mathematics is ‘fictionally correct’ just in case it would be true in the story of mathematics told by the platonists, were there abstract objects such as the platonists take mathematics to be truly about.

Sparse vs plenetudinous versions of this….Balaguer favors plenitude, because it avoids The Benacerraf Problem.

Will argue that T-fictionalism’s prospects for dealing with the issue of mathematical objectivity are just as good as – indeed tied to – the platonist’s prospects.

Platonism and Objectivity

Silly Platonism:  theorems about all non-isomorphic platonic structures are true per se. 

To avoid this, we need the distinction between true in a structure and true per se.

Better Platonism: To be true per se a pure mathematical sentence needs to be true in the intended structure, or intended part of the mathematical realm, i.e., the part we have in mind in the given branch of math.  Such intentions are captured by the ‘Full Conception’ (‘FC’) that we have of the objects (up to isomorphism).

FCs: to be captured by sentences.  But sometimes our intentions are just committed to whatever the sentences say; while other times they are committed to the sentences on the proviso that they capture our background pre-theoretic intuitions.  (e.g., Group theory vs. Natural Number Theory)

FCNN picks out an unique structure up to isomorphism; the one that we intend (contra Putman in “Models and Reality”, he says with conviction)

FCUS: Does it pick out an unique structure up to isomorphism? Field thinks not, but there is more to be said, as we’ll see.

Infinitely Better Platonism (IBP):  A sentence S of pure mathematics is true iff true in all the parts of the mathematical realm that count as intended in the given branch of mathematics, and there is at least one such; and S is false iff false in all such parts of the mathematical realm (or there is no such part); and if S is true in some intended parts of the mathematical realm and false in other intended parts, then there is no fact of the matter as to whether it is true or false.

Ruling out the possibility of indeterminacy would require sparse platonism, but that would make mathematical truth arbitrary and mathematical discovery impossible.  What grounds could we have for choosing one of the non-isomorphic structures over the other, barring some new addition to our FC?

IBP is non-revisionistic wrt external constraints on mathematics. 

‘Infinitely Better Fictionalism’ (IBF), the best version of T-fictionalism:  A pure mathematical sentence S is correct iff, in the story of mathematics, S is true in all the parts of the mathematical realm that count as intended in the given branch of mathematics; and S is incorrect iff, in the story of mathematics, S is false in all intended parts of the mathematical realm; and if, the story of mathematics, S is true in some intended parts of the mathematical realm and false in others, then there is no fact of the matter as to whether S is correct or incorrect.

But how does this preserve mathematical objectivity that extends beyond our current theories?  Here is how: (see pp. 150-151).  The IBF Fictionalist will ape the IBP Platonist.  Apparently, the IBP Platonist can say wrt some newly intuited axiom A, which together with ZF implies the Axiom of Choice, that its intuitivity shows that it was already part of our FCUS, we just hadn’t noticed.  The IBF factionalist can just mimic that.

For the IBF fictionalist, correctness facts  stolen from the Platonist will decompose into empirical facts about what exactly is built into our intentions, plus logical facts – consistency and entailment facts about what follows from the supposition that there exist mathematical objects of the kind that correspond to our intentions, and our FCs, and the axiom systems we work with.  So correctness facts decompose into logical facts of this type plus “an empirical residue”.  So mathematics is to that extent empirical.

From David Papineau’s Philosophical Naturalism

‘Fictionalism’ wrt a ‘domain of discourse’:

1.     A literal understanding of the claims made about the domain.

2.     A rejection of belief in such claims.

3.     An acceptance of such claims as fictions which are useful for various pragmatic purposes.

Versus Realism on the one hand and Instrumentalism on the other.

Examples of factionalist construals.

Mathematics:  One should not believe that 2+2=4.  Rather, one should accept it, on its Platonist construal, as a useful fiction.  E.g., what it implies about particular concrete countings is true and believable, such as that adding these two chairs to those two different chairs will make a total of 4 chairs.

Morality:  One should not believe that murder is wrong.  Rather, one should accept that it is, as an expression of our impartial disapproval of murder.

Modality:  one should not believe that ‘Necessarily, if both P and (if P then Q), then Q’.  Rather, one should accept it as an expression of our unqualified commitment to corresponding instances of argument having that form.