The main problem of scientific underestimation is reduced to the idea that evidence is not enough to determine our beliefs in a scientific theory. However, evidence based on empirical observations does not always insure the correctness of theoretical concepts because it can reject what was determined to be correct before.
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Quine’s holism rejects “the distinction between empirical and a priori truths, where a priori truths are those that are known independently of experience and immune to revision in the light of it” (Shapiro, 2005, p. 416). The interpretation of the statement shows that epistemic beliefs are included into the network of epistemologically important hypotheses connected with other background beliefs. However, as hypothesis is based on empirical implications or consequences, a falsified consequence can lead to wrong explanation for related background assumptions. Additionally, contrastive underestimation constitutes that a body of evidence confirming one theory can confirm other theories as well.
The indispensability argument put forward by Quine and Putnam relies on the view that scientific theories are justified or unjustified as wholes (Quine, 1980, p. 41). Based on the above, one has to concentrate on the issues of epistemic knowledge and the challenges created by the empirical observations. Further, one should discuss various opinions and arguments for and against holism to identify the place of this theory in network of beliefs. Finally, nature of mathematical knowledge as applied to different scientific theories should be examined.
The indispensability argument to science provides a solid ground to believe in the existence of mathematical objects, like sets, numbers, and functions. This argument is premised on the view that if a scientific theory is justified by empirical implications, then the entire theoretical framework is justified as well. The view is called confirmational holism. However, there is an ambiguity concerning the basis of holistic justification. Empirical findings cannot fully guarantee the indispensability of Mathematics to science because the obviousness of mathematical objects is left unconsidered by Quine.
Confirmational Holism, Its Main Concepts, and Application to the Indispensability Argument
Overview of Indispensability Argument
According to Quine (1990), our beliefs relate to the ultimate evidence, and this evidence is directly connected with the whole system of beliefs but not with its isolated elements. The above implies that the evidence for particular object existence should not be explicit but derived from our own system of beliefs (Shapiro, 2005 p. 413). Consequently, one ought to define the criteria for outlining the objects committed to our general system of beliefs. Presenting science as the result of development of evidence-based system of statements, Quine considers it as the external arbiter of truth (Colyvan, 2003, p. 11).
In this respect, Mathematics is considered to be an inherent part of science whose significance matters in correlation with other scientific theories (Shapiro, 2005, p. 419). The ontological commitment of mathematical components to science requires the recognition of the existence of the mathematical entities. The problem presented by Quine has no mathematical evidence; instead, he accepts the possibility for ongoing revisions of beliefs. Consequently, one can harbor misconception for the original system of mathematical objects creating ontological relativity.
To pursue this relative tendency, it is reasonable to consider the connection between empirical evidence and scientific theories. Therefore, it is important to consider Quine/Putnam indispensability argument (Colyvan, 2003, p. 11). This will explain the genesis of objects and our beliefs as well as the correlation between them.
Indispensability argument proposed by Quine and Putnam deserves considerable attention because it has been acknowledged as one of the best reasoning for mathematical realism (Platonism). Pertaining to mathematics and its integral connection to science, Quine states, “[o]rdinary interpreted scientific discourse is as irredeemably committed to abstract objects – to nations, species, numbers, functions, sets – as it applies to other bodies. … The numbers and functions contribute as genuinely to physical theory as do hypothetical particles” (Quine, 1981, pp. 149-150 as cited in Colyvan, 2003, pp. 9-10).
The above argument seemingly addresses the overall holistic vision on the metaphysical state of affairs. Calling for ontological commitment, Quine fails to explain the first premise of the indispensability argument that supports the second one. There is little evidence proving the ontological ground for the indispensability of all entities to the best scientific theories. This is why it extremely needs alternative explanations and defense.
For many supporters of Platonism, the indispensability argument for the presence of mathematical entities provides the best evidence for mathematic realism (Stanford, 2001; Colyvan, 2003). Nevertheless, it has undergone serious attacks for the reason that this argument, particularly its conclusion, does not fit well mathematical practice. Therefore, mathematics can be considered to be indispensible in improper way. Taking into account Colyvan’s intention to compare Quinean’s appraisal of unapplied mathematics with ‘recreational’, it suggests that if one considers the problems from Quine’s naturalistic approach, one should infer that all mathematical aspects are recreational as well. In other words, science appeals to Mathematics due to its efficacy.
The central idea of Quine’s confirmational or epistemic holism suggests that our knowledge is the web of logically connected beliefs (Colyvan, 2003, p. 34). Therefore, when we produce a hypothesis about some event or phenomenon, we attribute our entire system of beliefs to sensory experience but just to a part of it. There are different formulations of confirmational holism and one of them sounds as follows:
The falsity of the observation categorical does not conclusively refute the hypothesis. What it refutes is the conjunction of sentences that was needed to imply the observation categorical. In order to retract that conjunction we do not have to retract the hypothesis in question; we could retract some other sentence of the conjunction instead. This is the important insight called holism (Quine, 1990, p. 13).
Certainly, the observations carried out can definitely contribute to the creation of new theoretical concepts. However, the removal of false statement does not guarantee that the final concept will be correct due to the implicit ground of the original system of beliefs. Aside from this, if beliefs are justified, the ontological commitment still allows to admit that there exists disconfirming evidence beyond the sensory experience. In other words, Quinean statement turns out to be self-refuting thus including a portion of skepticism into this position.
Considering the existence of mathematical objects as the set of hypotheses and beliefs, one cannot rely on the empirical observations only, excluding the analytical and synthetics procedures. Holistic approach admits that if Mathematics is justified empirically by the best scientific theories, all the mathematical entities are justified. The argument would be irresistible but for the probability of existence other theories and entities which have not been discovered yet.
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Therefore, it is reasonable to take the science as the purpose for mathematical components but no more that (Colyvan, 2003, p. 11). There is a variety of scientific goals where it is possible to build up a number of indispensability arguments. In this respect, science considers it necessary and convenient to use mathematical notions since Mathematics acquires significance only when it is applied to science.
Holism vs. Naturalism
The first premise of the indispensability argument can be interpreted as follows: “we ought to have ontological commitment to all and only the entities that indispensible to our best scientific theories” (Colyvan, 2003, p. 11). However, this premise is in need of support that comes from tenets of naturalism and holism.
According to Quine, natural science serves as a final arbiter of truth and existence. This is the main thesis of naturalism revealing that Mathematics is true and the presence of mathematical entities is as well justified as that of the other objects set by science. The grounds of holism provide no substance without holistic explanation.
As it has been investigated previously, confirmational holism suggests that observational evidence for natural science relates to the theoretical framework as a whole rather than to individual component hypothesis. Together with naturalism, holism provides the justification for the indispensability argument. In whole, naturalism provides the “only” and holism provides the “all”, as presented in the first premise.
Quine’s position reveals two aspect of holism. The first one is confirmational holism as presented above and the other one is semantic holism suggesting that meaning cannot be expressed by a single sentence but by system of sentences. These holistic themes are closely related but still there are many reasons to distinguish between them. More importantly, most authors (Resnik, 1999; Maddy, 2007) believe that confirmational holism presents a bulk of controversies, particularly in the context of Quine’s indispensability argument.
Key points of Holistic Underestimation
Quine (1990) argues that a single hypothesis cannot carry any information on the possibility to contemplate in nature (p.13). One would rather draw empirical products from this single hypothesis in case it conforms to other beliefs, rather than include background insights about the world. Such empirical findings will also be true in case it is logically connected with further hypotheses derived from the interaction between objects, as it is presented in the original assumption. In case empirical prediction appears to be false, there is no chance to define whether the falsity lies in the original hypothesis or in its wrong connection with the entire system of beliefs (Duhem, 1991).
Quine’s position, however, does not argue the existence of recalcitrant experiences; it disputes how those experiences can influence the formation of the original system of beliefs. Thus, a problem arises concerning the validity of this system since some of assumptions may turn out to be inadequate. Here, Quine (1990) insists on the idea that such system is possible to preserve and to follow in case sufficiently solid justifications are provided in different fields of the web (p. 15).
Problems of Holistic Justification with Regard to Epistemology
Revealing the epistemic aspect of indispensability, one should refer to two main streams in the philosophy of mathematics presented through realistic and ant-realistic views. Both positions unveil their own understanding of such mathematical entities as numbers, functions, and sets. The cognition of those entities from two points of view is predetermined by their relation to the material world and to human knowledge in particular (Covylan, 2003, p. 2).
Based on the above deliberations, mathematics is comprehended through different perceptions of the physical world and its connection to logic. It means that “mathematical entities such as functions, numbers, and sets have mind- and language-independent existence or, as it is also commonly expressed, we discover rather than invent mathematical theories” (Colyvan, 2003, p. 2). Such an explanation definitely takes its roots from the metaphysical realism. This statement is opposed to anti-realistic views on mathematics. The supporters of this school of thought are inclined to think that mathematical theories are invented rather than discovered by a human mind.
The realistic position suggests that mathematic components are recognized via truth. Truth exists apart from our knowledge and it is not necessary to refer to metaphysics to define whether the object is true of false. This theory of objectivity does not contribute to explanation for the existence of mathematical entities (Wray, 2002, p. 307). Therefore, the question of indispensability is attributed to metaphysic explanation as far as the nature of mathematics is concerned.
The significance of metaphysic realism, or Platonism, lies in setting mathematics into the physical realm and in proving that mathematical knowledge can co-exist with other branches of knowledge (Colyvan, 2003, p. 5). It reveals a complicated relationship between empirical science and epistemology that contradict one another. This is why Quine’s argument is premised on several important assumptions where the first one – conformational holism – discovers that only essential bodies of theories can be empirically tested.
The main problem here addresses the relation of those theories to observational evidence since a mere dependence on sensory ‘input’ is not enough to reconstruct a theoretical vocabulary. Additionally, as the notion of indispensability is consistent with the notion of a priori truth, the existence of mathematical object should be absolute as well. However, the existing evidence can reveal only one dimension of this justification since the notion that is approved unilaterally may fail when passing different testing (Wray, 2002, 308).
Objection to Holism: Debates
There have been many objections to the indispensability argument, particularly to confirmational holism. In particular, the controversies are based on the claims that empirical science cannot serve as a confirmation of mathematics (Sober, 1993; Maddy, 2007; Shapiro, 2005). Therewith, there is a presumption put forward by Maddy that confirmation is based on the use of mathematics as a useful instrument for creating theoretical concepts with no explanation of this application (Maddy, 2007, p. 316). All these positions reveal reasonable objections to Quinean holism that also imposes criticism on the indispensability argument.
There are different views on the validity of the holistic justification. In response to holism, there exist alternative theories, like the atomic theory, revealing that relevant scientific method does not consider the existence of all statements of scientific theories as justified (Shapiro, 2005, p. 441). Apparently, the majority of mathematical objects appear in extra-empirical situations when physical phenomena are materialized in the form of geometrical formulas, or “when large finite collections are treated as infinite, when discrete connections are treated as continuous” (Shapiro, 2005, p. 412).
Pertaining to atomic theory, the evidence is defined as atoms of physical objects identified by theoretical virtues. It provides a solid ground and explanation of the indispensability argument (Shapiro, 2005, p. 441). However, if the evidence of the atoms is more explicit, since they are detected as medium sized physical objects, then, the nature of mathematical objects is not the same.
When deliberating on the actual reason of application of mathematical objects to science, one can encounter with such an explanation as effectiveness and convenience. Therefore, if theories apply for these entities, the success of the entire theory may be under the threat. Relying on this analysis, the indispensability argument cannot be justified since the current scientific standards do not confirm all parts of scientific theory and, more importantly, the mathematical objects are excluded from the justified statements (Shapiro, 2005, p. 455).
A closer consideration of Quinean holism, specifically the confirmational model, shows that the majority of philosophers are inclined to consider this position to be implicit (Resnik, 1999, 118; Shapiro, 2005, p. 419; Maddy, 2007, p. 315). In particular, Quine narrows the set theories to their capacity to produce standard theoretical models and foundations for mathematics cutting down the contradictions (Quine, 1990, p. 16). Still, if the existence of mathematical objects is irrefutable, mathematical existence as itself is under the question.
Apart from this, the main disadvantage of Quine’s theory consists in its failure to revise the fixed notions of the mathematical theory in the light of empirical findings (Shapiro, 2005, p. 419). However, if the concepts are not revised, it does not indicate the impossibility of revision. In that regard, even if observation disapproves the prediction, there is still a chance for revising the auxiliary assumptions. The problem is that well-grounded methodology does not recommend adjusting to secondary presumptions. Quine’s own position, thus, it is too ambiguous to present a unique outcome.
Another objection to holism provokes a disputable ground with regard to mathematics and logic that is sustained by our scientific and mathematical experience. Pertaining to these intuitions, “mathematics and logic are fixed points in our investigation of the world, determining the limits of what we can entertain as serious possibilities” (Resnik, 1999, p. 118). The above refuses the idea of existence of mathematical evidence thus contributing to its ambiguous relation to scientific theory. Besides, the holistic presentation of scientific theory provides no ground for specific types of mathematical evidence. Consequently, empiricism can be premised on certain hypothesis, but not the entire systems.
In return to attacks on holistic grounds and the so-called ‘good sense’ as the only substantial explanation for the relationship between theoretical inheritance and observational evidence, many philosophers try to put forward their own explanations and objections to it (Shapiro, 2005; Maddy, 2007; Resnik, 1999). Indeed, the existence of independent judgments as the only evidence of the relationships between sensory experience and theoretical statements is ungrounded.
As an alternative, Resnik’s vision of ‘good sense’, as a sort of pragmatic rationality, insures a special significance of mathematics for science and presents mathematics as a priori. In order to explain the connection between Mathematics, scientific theories and the evidence, Resnik (1999) premises on the assumption that a combination of mathematics and science is created to produce concepts and models allowing to calculate the values and quantities concerned (p. 119). This assumption seems to be more winning as compared with Quine’s explanation.
The emphasis on the significance of mathematics for science due to its efficiency and convenience is a more relevant explanation for the relation between the two. Hence, it is impossible to omit the idea that mathematical models are closely associated to logical considerations rather than to individual hypotheses. Therefore, in order to conclude that evidence is based on a particular hypothesis, one has to provide a range of other assumptions as well as different variants of their idealizations. The justification of this mathematical model leads to pragmatic consideration, including mathematical tractability and theoretical simplicity. The mathematical truth, hence, is revealed only if it is applied to the construction of scientific models.
Epistemology vs. Empirical Evidence within the Discussion of Holism
As it has been informed earlier, one cannot but refer to the confrontation between empirical evidence and epistemology when discussing the role of evidence in the theory justification. There is also the assumption that it is possible to justify and to retain hypothesis based on the evidence (Laudan, 1990, p. 270). However, no empirical evidence can be convincing because it can reconcile all the constraints of theory and, at the same time, it can disguise other alternative possibilities for justification. Therefore, according to Laudan (1990), different variations, or auxiliary assumptions, presented in holistic theory cannot be justified since the empirical evidence supporting theory has different rates of validity.
Various dimensions of holistic underestimation reveal that various responses to unconfirmed evidence are logically explained rather than subjected to rational defensibility. This premise appeals to further epistemic studies revealing the idea that extra-empirical evidence is insufficient for narrowing the logical possibilities of the beliefs that are rationally accepted. Considering Quine’s holistic position, Laudan (1990) denies the possibility of logical and even psychological revision of beliefs believing that if the primary question is unknown, it is impossible to make further assumptions as they may turn out to be false (p. 270). The statement is also approved by other schools of thought (Maddy, 2007; Shapiro, 2005; Resnik, 1999).
As a proof for everything mentioned above, a solid explanation is provided for the rejection of empirical evidence as the problematic notion that has no safe application (Laudan 1990). It is suggested that the idea that “the empirical equivalence of group of rival theories, should it obtain, would not by itself establish that they are underdetermined by the evidence” (Laudan and Leplin, 1991, p. 450). Therewith, the arbitrary pluralism of beliefs applied to theories contributes to epistemological skepticism.
The point is that theories can be empirically equivalent if they have the same level of empirical consequences. In its turn, empirical consequences have their own level of equivalence that is based on observations and logical assumptions. In case the original hypothesis is not justified, the entire theory cannot be confirmed as well as subsequent chain hypotheses will be misleading. The same pertains to Quine’s holistic view on mathematical theory where the existence of mathematical objects is subjected to ambiguity concerning its originality.
Certainly, the application of empirical evidence for outlining and creating new theoretical knowledge remains justified but partially and, therefore, it cannot constitute a serious problem for holistic underestimation. Hence, empirically equivalent theories are often perceived as available alternatives in case there are no other supportive arguments (Standford, 2001, p. S3). At this point, Quine’s holistic approach to indispensability argument should be studied, as there is a chance that one auxiliary assumption can be justified within other theoretical framework. Despite the existing controversies, this argument is still applied in many theoretical frameworks.
The indispensability argument revealing the significance of Mathematics for science has certain controversies due to its empirical ground. After a thorough consideration of all the positions against the conformational holism as the basis of the argument, the protests are directed at the contradicting connection between theoretical statements and sensory experiences where Quine’s ‘good sense’ has been subject to merciless attacks.
The conformational holism cannot serve as a reliable ground for creating the chain of beliefs where one of them may turn out to be inadequate thus rejecting what was assumed before. Apart from this, confirmational holism has also provided positive contributions to the indispensability argument as well as to empirical science. The holistic approach, therefore, serves as the key for justifying practical side of scientific theory.
In general, the main problems of holistic justification refer, first of all, to the ambiguity of the original empirical findings and further logical deductions from them. Second, holism creates problems during the interaction of different branches of scientific knowledge since the introduction of local changes to one theory is not available to another related theory. Third, extra-empirical evidence and extended system of beliefs contradicts the principles of epistemic knowledge when everything is learnt through sensory experience.
Finally, ‘good sense’ can lead to rejection of hypothesis because it can be based on a failed model or wrong assumption. Pertaining to Mathematics, the basis of indispensability argument is the existence of empirical observation, the basis of which is auxiliary assumptions and suggested chain of beliefs.
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