An Objection to the Quine-Putnam Indispensability Argument

1. Introduction

1.1. Platonism and Fictionalism

What are numbers? Do they exist?

Despite mathematics’s long-standing role in human civilization, these seemingly simple ontological questions remain unsolved.

Regarding the ontology of mathematics, diverse viewpoints exist, but they can generally be classified into platonism and anti-platonism. Mathematical platonism is the view that (i) abstract mathematical objects exist and exist outside of space and time (i.e., they are not spatiotemporal), and (ii) our mathematical sentences describe these objects. On the other hand, anti-plationism encompasses views that reject platonism from various perspectives. Among these, fictionalism is considered one of the most compelling. Like platonism, fictionalism points out that mathematical sentences describe mathematical entities; however, unlike platonism, it denies the existence of such abstract entities. From a fictionalist’s perspective, therefore, the statement “3 is prime” is false because “3” does not exist (Balaguer, “Fictionalism in the Philosophy”). In the following paragraphs, I will defend the fictionalism arguments against platonism in the context of the Quine-Putnam Indispensability Argument.

1.2. The Quine-Putnam Indispensability Argument

The most powerful support for mathematical platonism is often considered to be the Quine-Putnam Indispensability Argument. It states that (a) we must ontologically commit to entities that are indispensable to our best scientific theories, (b) mathematical entities are indispensable to our best scientific theories, and (c) therefore, we must ontologically commit to mathematical entities (Colyvan). For example, quantum mechanics (QM) is considered one of the most successful theories in the history of physics, so following the indispensability argument: (a) we must ontologically commit to entities indispensable in QM; (b) mathematical entities like Hilbert space and complex numbers are indispensable to QM; and (c) therefore, we must ontologically commit to entities like Hilbert space and complex numbers. The indispensability argument, in this form, is valid; if its premises are accepted, the conclusion provides strong ontological support for platonism. Therefore, for anti-platonist positions such as fictionalism, it is necessary to show that the Quine-Putnam Indispensability Argument does not succeed in establishing ontological commitment to mathematical entities.

There are two main types of objections to the indispensability argument. The first, exemplified by Hartry Field, attempts to show that mathematical entities are dispensable in some empirical theories. However, this argument is problematic in many ways. For instance, many argue that Field’s nominalization does not fully eliminate mathematics from empirical theories. Moreover, for his argument to be universally valid, it would need to demonstrate that mathematics is dispensable for all representative empirical theories - a task that is both complex and burdensome. For these reasons, this essay will focus on the second type of objection: accepting premise (b) (that mathematical entities are indispensable to science) while challenging premise (a) (that we must ontologically commit to indispensable entities).

To clarify, this essay will adopt an ontological stance aligned with fictionalism, but the majority of the discussion will focus on objections to premise (a). Specifically, I will argue that many aspects of the nature of science conflict with premise (a). Starting from Section 6, the essay will shift to a defense of fictionalism. A common misconception here is that fictionalists must account for indispensability itself. However, the real challenge lies in explaining the relationship between mathematical theories and scientific theories under the assumption of indispensability (Platonism and Anti-Platonism 93–126). In Section 6, I will show that while platonists assert that anti-platonist views fail to account for this indispensable relationship between mathematics and science, platonist views also have trouble explaining this relationship when considering the nature of science as I have characterized it. Thus, the problem posed by indispensability cannot be attributed solely to fictionalism but also extends to platonism. Furthermore, in Section 7, I will go through a unique objection raised by fictionalists regarding the metaphorical use of mathematical language and suggest strategies that fictionalists might use to address the challenges posed by indispensability.

2. Review of Premise (a)

2.1. Naturalism and Confirmational Holism

Before questioning premise (a), it is important to first identify the ground on which it rests. Premise (a) is supported by the combination of naturalism and confirmational holism (while Quine uses other forms of holism, confirmational holism is considered the only necessary component for the indispensability argument).

Naturalism provides support for the ontological belief of entities in our best scientific theory by stating that science, with philosophy as a continuous part (they work together and are non-separable in naturalistic inquiry), can provide a complete picture of the world. Confirmational holism, on the other hand, holds that theories are confirmed or disconfirmed as wholes rather than in separation. In other words, naturalism offers the justification for treating our best scientific theory as the only source of ontological commitment (as indicated by “best”), while confirmational holism ensures that all components of a confirmed theory (as in “all entities”), including its mathematical elements, are equally validated, as implied in premise (a). Together, these two theories imply that if our best scientific theory is confirmed, then its mathematical components are also confirmed (Colyvan).

2.2. The Scientific Practice Objection

One of the strongest objections to the conjunction of naturalism and confirmational holism is proposed by Penelope Maddy. Maddy argues that confirmational holism is untenable because scientists in practice do not treat all components of a well-confirmed theory with equal epistemic confidence. For example, despite the empirical success of atomic theory in the late 19th century, many physicists remained skeptical about the existence of atoms until Einstein’s 1905 work on Brownian motion. This historical example indicates that empirical confirmation may validate a theory’s predictive success without necessarily warranting ontological commitment to all its theoretical entities. In this case, Maddy shows that, from a naturalism perspective, confirmational holism overstates the epistemic warrant for ontological commitments to elements in our best scientific theories.

Furthermore, Maddy anticipates and rebuts a potential platonist response to her critique. A platonist might argue that the mathematical components of theories always fall within true elements rather than merely useful elements. However, Maddy counters this by pointing out idealizations in scientific theories—such as modeling water as infinitely deep in wave theories—which use literally false assumptions. These cases demonstrate that mathematics often functions as an instrumental tool for idealized modeling rather than as a descriptive truth-teller. Consequently, this platonist view fails because it contradicts scientists’ actual use of mathematics in theory construction (Maddy 275-289).

In conclusion, through science’s historical practices, Maddy shows that the empirical success of a scientific theory cannot provide sufficient epistemic support for the ontological commitment of its constituents. Also, she argues that mathematics has been used more as an instrumental tool rather than as a description of truths. Therefore, the empirical success of a scientific theory cannot lead to the conclusion of ontological commitment to its mathematical parts.

3. Objection 1: Against Confirmational Holism

In this section, I will dissect Maddy’s objection against confirmational holism and develop her theory with a focus on how confirmational holism contradicts some aspects of the nature of scientific inquiry. To recall, confirmational holism states that theories are confirmed or disconfirmed as wholes. This theory appears plausible on the disconfirmed part: if a theory is disconfirmed or contradicts our observations, then it is possible for any of its hypotheses, auxiliary assumptions, and background knowledge to go wrong. However, when reversing the condition to a theory being confirmed, it is still possible for some parts of it to go wrong. This problem of holism comes from mistakenly treating “confirmed” or “disconfirmed” as an analogy of “sound” or “unsound” in a statement. When we say a statement is sound, its premises, logic chain, and conclusions are all true, but this is not the case of “confirmation” in science. This analogy, in essence, is the problem of treating science as a perfectly or completely objective discipline.

3.1. The Aspects of the Nature of Science

Actually, our science is not completely objective or limited to descriptions of objective natural laws, and we do not judge science only on the basis of its accuracy in describing the world (scientific theories do not usually function as the most accurate description either). In fact, our scientific theories carry our aesthetics and perspectives and meet our requirements, which can be shown in various ways.

Aesthetically, modern scientists tend to believe the simpler and more beautiful equations are more likely to be true, and the Occam’s Razor mindset is common in science. Einstein’s theory of relativity, for instance, was praised for its elegance and accepted rather easily due to this elegance. Eddington, an eclipse project implementer, was already convinced by Einstein’s theory before making observations (Brush 184-214). This indicates how the acceptance of a theory does not solely depend on its objectivity or accuracy in accounting for reality.

Socially, the choices of theories are also dependent on the social and cultural background. New theories that successfully replaced old theories have usually occurred in similar forms before (Kuhn 66-110). However, those theories had not been accepted until then because the cultural or social background before was not the most suitable for them. For example, although Copernicus was usually known as the famous advocate of the heliocentric model, Aristarchus of Samos had proposed a heliocentric model nearly 2000 years before Copernicus, but his model was ignored for centuries. This was partly because the geocentric model fit better with the religious beliefs. Thus, sometimes it is the social and cultural changes that drive attention to certain scientific theories or the intention to change scientific theories. This further shows that both the construction and acceptance of scientific theories are dependent on other factors than objectivity, like our social and cultural backgrounds.

Therefore, the nature of our scientific theories has at least two aspects: (1) it describes the physical world; (2) it carries the characteristics or fulfills the requirements of humans.

3.2. Contradiction with Confirmational Holism

After stating the aspects of the nature of science, we can now specify that confirmational holism is mistaken because it ignores (2), and there will always be errors and simplifications in our scientific studies. To show this clearly, imagine:

Scientist A wants to test Ohm’s law. I found the best batteries, resistors, ammeters, voltmeters, and wires in the world, meaning that the battery has very low internal resistance; the resistor’s resistance almost stays constant through the experiment; the ammeter has almost zero resistance; the voltmeter has almost infinite resistance; and the wire has almost zero resistance. Through experiment, he found a fluctuation of magnitude V around the assumed linear relationship. If A treats “confirmed” or “disconfirmed” as strictly “true” or “false” in this case, then A should disconfirm Ohm’s law because it is different from the real-life situation. In this case, confirmational holism will be correct because we can question anything, like “There is no ohmic resistor.” However, anyone with a little knowledge of science would choose to confirm the theory because the error is acceptable for humans. In this case, the theory is confirmed, but there is still no such thing as perfectly ohmic material.

Therefore, scientific theories are descriptions of the world after different processes by humans. When processing, such as making simplifications, we will definitely do something that is not strictly supported by the physical world, i.e., an ohmic conductor. Thus, whether a scientific theory is confirmed or not does not solely depend on the accuracy of results but also on other characteristics, including but not limited to acceptance of assumptions, simplicity, aesthetics, etc. It is the reasonable simplifications and methodologies chosen by humans and the acceptable conclusions that make a theory confirmed. These humanistic characteristics give science a new standard of confirmation and structure that is essentially different from what soundness is and the structure it implies for an argument. This structural difference makes the analogy to soundness required in confirmational holism wrong, which disables premise (a) and therefore the whole indispensability argument.

3.3. Conclusion

In conclusion, confirmational holism is mistaken because it ignores aspect (2) of science’s nature. Due to aesthetic-based, convenience-based, and other humanistic demands of science, theories like Ohm’s law can be confirmed even if they make literally false simplifications, like the resistor is made of ohmic material.

4. Objection 2: Definition of “Best”

In this section, I will extend that premise (a) faces the problem that it is hard to define “best” in science, and this problem, in essence, is still the result of ignoring aspect (2) of science’s nature.

To start with, it is rather easier to show what the adjective “best” does not mean in scientific studies—it definitely does not mean the most accurate theory. All of the examples I gave in this essay can show this. For example, if science is purely for accuracy, then Eddington would not accept Einstein’s theory before making observations, and the concept “ohmic resistor” would not exist at all. We have different requirements when using scientific theories, and the judgment of scientific theories should be put into context.

For example, Einstein’s relativity is considered more accurate than Newtonian physics, especially in extreme regimes like high speeds. However, we still frequently use Newtonian physics in many of our daily life problems now (i.e., engineering, sports analysis, automobiles, etc.) because it is simpler and easier for computation. In these situations, Newtonian physics is actually better than relativity since it fulfills our requirement of convenience better without much of a loss in accuracy.

This causes a problem for the indispensability argument because we cannot identify which scientific theory is better without context. Additionally, we cannot accept both simultaneously from a scientific standpoint because they are contradictory in many respects. Believing in both at the same time, therefore, would be inconsistent. Someone might say that we can classify the situation into different contexts and accept the one according to our classification. Nevertheless, this is in conflict with the principle of causal isolation (PCI). Commonly accepted by platonists, PCI states that there are no causal interactions between mathematical and physical objects (Balaguer, Platonism and Anti-Platonism 93–126). If we consider the “best” theory relatively (according to the situation), then we have to ontologically commit to mathematical entities according to our classification of situations. This is in conflict with PCI because the ontology of mathematical entities, in this case, is causally related to real-life conditions. In other words, when we walk from one place to another, we might have to ontologically commit to different mathematical entities, which is not reasonable at all.

5. Objection 3: A Historic View

In this section, I will show that premise (a) faces the challenge of instability due to another aspect of the nature of science when viewing its history: our best scientific theories are constantly changing (3). This aspect can be easily justified and shown in various cases.

5.1. An Example of Luminiferous Ether

For a long time in history, physicists believed that electromagnetic waves like light need a medium for transmission, and luminiferous ether was assumed to be the medium (Gregersen). Luminiferous ether was commonly assumed to be weightless, transparent, frictionless, undetectable chemically or physically, and literally permeating all matter and space (there were debates on the exact nature of luminiferous ether). This theory was commonly accepted at that time, and physicists at that time usually used elastic solid equations to describe luminiferous ether (Leach and Sher 4). However, after general relativity is accepted, the theory of luminiferous ether is immediately discarded because it is no longer necessary, and tensor calculus is introduced. The replacement of luminiferous ether shows that our scientific theories are evolving dynamically, so do the mathematical entities used in them.

5.2. Contradiction with the Indispensability Argument

According to Quine, he would only ontologically commit to the mathematical entities that are necessary in our best scientific theories, while other mathematical entities would only be considered as mathematical recreation without ontological right (Quine). However, if we trace science dynamically, we can find out that due to the frequent changes in our best scientific theories, the mathematical entities in those theories have been frequently discarded. Therefore, according to Quine, when the relevant theories are rejected, we will inevitably admit the existence of some mathematical entities at some point and categorize them as mathematical recreation.

This will cause problems for two main reasons. First, we cannot correctly identify or make sure of the existence of basically any mathematical entity. Even if we assume the whole discipline of mathematics is indispensable to our science, there is a possibility for each mathematical entity to be replaced by another. This means when taking a sporadic view, we do not have the confidence to commit to each individual mathematical entity, which makes the indispensability argument meaningless. Second, this will cause inconsistency with PCI. Following the indispensability argument, if we use science to decide what mathematical entities we should ontologically commit to (or classify their importance—existence or recreation), our scientific theories would have a causal relationship with our ontological commitment to those mathematical entities, which contradicts PCI. Therefore, even for a Platonist that agrees with PCI, the indispensability argument would not be able to provide satisfactory methods of ontologically committing mathematical entities.

A possible misunderstanding or weak objection towards this view is that our core scientific theories are not changing, implying the mathematical entities behind them are also not changing. Even though those theories are more likely to be stable, most of them still went through significant changes. Atomic theories, for example, are fundamental theories in chemistry that have developed from Bohr’s planetary model, Dalton’s solid sphere model, Thompson’s plum pudding model, Rutherford’s nuclear model, and Democritus’ atomism to Schrödinger’s quantum model. In this process, several mathematical frameworks were introduced and discarded.

A better objection would be that many of our mathematical entities remain even though scientific theories are constantly changing, but this is only an overconfidence in our mathematics. Euclidean geometry was once considered as one of the most indispensable mathematical systems in science; however, it was replaced by Riemannian geometry in general relativity. Furthermore, the relatively stable mathematical entities are stable for other reasons. such as pragmatic ones, rather than ontological ones. This can be shown because we actually revise mathematical entities. As Field points out, examples like replacing infinitesimals with limits in calculus show that the original ontology is not important when compared to utility (Field).

5.3. Conclusion

In conclusion, aspect (3) of science’s nature of frequently changing makes it almost impossible for humans to distinguish between practical mathematical entities and mathematical recreation, or between existence, if there is any, and non-existence in mathematics. As a result, the indispensability argument becomes problematic, since as an argument that informs people about ontological commitment, it could not provide strong enough confidence for people to believe in any of their mathematical entities.

6. Problem of Platonism under Indispensability

To recall, fictionalists do not need to account for the indispensability itself. Rather, they need to provide explanations of the relationship between science and mathematics under indispensability. In this section, I will show that with the three aspects of the nature of science that I provided, platonists will also need to provide such explanations, and it would also be problematic for them.

Following from the previous objections, our scientific theories do at least three things: (1) describe the world; (2) carry the characteristics or needs of humans; and (3) constantly change. The sense that the indispensability argument supports platonism only comes true under the misunderstanding that science only does (1), which pushes science to an objectivity level that it should not have, as it ignores the humanistic and dynamic parts of our best scientific theories. When considering both, the indispensability argument can also cause problems for platonists. Platonists usually question: if mathematical entities do not exist at all, why would mathematics function that indispensably for the science that describes the world? However, fictionalists can also question platonism based on aspects (2) and (3) of science’s nature. We can ask the following question regarding (2): if mathematical entities exist, why would mathematics be so essential to the science that possesses human traits? especially when it does not causally relate to science (according to PCI)? For (3), we can question if mathematical entities do exist non-spatiotemporally, why would they be so indispensable for the science that frequently changes through time?

6.1. A Case of Infinity

In objection 3, I already showed that platonists cannot arrive at a satisfactory explanation for the question regarding (3). Therefore, I would only show why they also struggle to explain the question regarding (2) in this subsection through the case of infinity.

Infinity is usually used as an assumption in empirical theories. For instance, the analysis of water waves usually assumes water as infinitely deep; the standard model of particle physics assumes the size of an electron as infinitely small, and general relativity assumes space as infinitely divisible. Obviously, there is no infinitely deep water in real life; electrons have their sizes, and there exists a Planck length, so we should account for these mathematical usages of infinity.

A platonist’s account of infinity might take a similar form to this: out of space and time, there exists something infinite, and those nonexistent infinities in our scientific theories are imperfect projections of this abstract infinity. However, this explanation will cause problems if we dive deeper into these assumptions. It seems that for reasons like simplicity, we choose to describe many physical entities in our theories as having infinite magnitudes, rather than because we really cannot distinguish their magnitudes with infinity (i.e., we can measure the depth of water and the size of an electron). In this case, it is very hard for platonists to explain why mathematical entities, which are causally isolated from physical entities or even humans, could function indispensably for subjective human choices.

7. Approaches that Fictionalists Might Take

In this section I would provide two approaches or general directions that, when combined together, fictionalists can use to explain why mathematical entities have such a high applicability in our scientific theories.

7.1. Metaphors

To start with, I will show a unique objection by fictionalists - metaphors. Yablo has first argued that mathematical sentences are advanced in a make-believe spirit (“Explanation” 1007-1029). Also, mathematical objects function metaphorically rather than in a literally true manner (“Go Figure” 72-102). This argument is intuitively sensible since when we say there are 8 planets in our solar system, we do not intend to refer to the non-spatiotemporal object “8”; rather, we are expressing it in a figurative way that deviates from the literal meaning. Similarly, sometimes we also say “the average woman,” while there doesn’t exist one. This shows the gap between indispensability and existence.

7.2. Structures

Let’s go back to the infinity case. A fictionalist can avoid the problem that platonists face by simply stating that infinity does not exist but is merely useful. This claim explains all three aspects of the nature of science that I stated, especially for (2) and (3). Also, it fits well with the objection through metaphors because it can explain metaphors in our science as based on pragmatic use, which avoids ontological commitment. By showing that mathematics is just a useful tool but not a rigid existence, we can say that mathematics functions that well in fulfilling our needs and adapts that well to our constantly changing scientific theories because its aim is to be a useful tool, but how are mathematics useful tools for science?

I believe Balaguer makes a point by saying that fictionalists can point out that mathematical entities are so applicable or helpful in our scientific theories not for ontological reasons, but because mathematics provides useful structures or frameworks to describe the physical world (Balaguer, “A Fictionalist Account”).

By implying this view, fictionalists can deal with questions like why real numbers are so useful in describing temperature. We can simply explain this by saying that our perception of temperature fits well with the structure of the real number continuum. It is not each individual real number itself that is not dispensable, because we have Fahrenheit, Celsius, and Kelvin (K). In this case, the same real number can represent different physical temperatures. Similarly, for the indispensable metaphor of 8 planets in the solar system, we can say that the structure of positive integers, not their existence, is paralleled with the number of planets from our perspective.

To clarify, by saying structure is the thing that matters, I am not taking the ontological view of some structuralists that structures exist non-spatiotemporally. Such a view falls into the same problem of explaining aspects (2) and (3) of science’s nature. Mathematical structures are important for us as they represent our perception of the world, but not simply an objective description of the world.

7.3. Human Created Mathematics

Following from above, a more reasonable statement should be that mathematical entities and structures are all human conventions. Humans create rules (axioms) through experiences and inductions and deduce under these rules to create systems of mathematics. This even better explains why mathematics functions that well for humanistic needs in science - mathematics is a human convention.

In the case of infinity, we can take Poincaré’s view that infinity is created due to human intuition. He states that we are using the same syllogism infinitely to mathematically induce infinity, and the fact that we believe the syllogism can be used infinitely many times comes from our intuition. In other words, we face syllogisms like this:

The theorem is true for number 1.

Now, if it is true for 1, it is also true for 2.

Therefore, it is true for 2.

Now, if it is true for 2, it is also true for 3.

Therefore, it is true for 3, and so on and so forth.

We get infinity when doing such syllogisms repeatedly, but logic cannot prove that this syllogism is still right after being applied infinitely many times. Therefore, infinity actually represents a tendency created by our intuition and mind (Poincaré 1-22). This perfectly represents how humans can shape mathematics and explains why mathematics functions that well for humans. In this case, we can consider the mathematical truths as man-made truths, considering them as true only when following the rules in their own “fiction.” Euclidean geometry, for example, could be considered as a human convention under human-created axioms and would only be true in the fiction where these axioms are met.

7.4. Conclusion

The objection to metaphors indicates that mathematics is a make-believe game by humans, and this leads to two explanations that fictionalists can take. First, regarding the applicability of mathematics, fictionalists can say that it comes from the structure of mathematics but not the existence of its individual objects. Second, regarding the nature of mathematics, fictionalists can argue that mathematics is a human convention. In this case, the fictionalists can gracefully fit into the three aspects of the nature of science I proposed.

8. Conclusion:

The essay basically consists of two parts: pure objections to the Indispensability Argument and objections from the fictionalism perspective.

For the first part, I objected to the premise that we must ontologically commit to entities that are indispensable to our best scientific theories. I have pointed out three aspects of the nature of science: (1) it describes the physical world; (2) it carries the characteristics or fulfills the requirements of humans; and (3) it is constantly changing. The sense that the Indispensability Argument makes sense and works for platonism mostly because it comes from neglecting (2) and (3). (2) will cause problems for confirmational holism as it shows that we choose to construct literally false entities or structures in our scientific theories for purposes like simplification, while I also showed that (3) will cause problems of instability in the application of the Indispensability Argument. Besides, I also pointed out the description of “best” of our scientific theories usually depends on situations, which creates conflicts with the non-spatiotemporal characteristic of mathematics that platonists assume.

For the second part, I showed there are also significant problems faced by platonism due to aspect (2) of science’s nature. Then, I gave a special objection from the fictionalists’ perspective and indicated possible approaches that fictionalists can take to explain the relationship between mathematics and science under indispensability: stating it is mathematical structures that function in science and stating that mathematical entities are human conventions.

However, in the second part, I only provided general directions that fictionalists can take. There are other possible problems not specifically mentioned in this essay that can be further addressed. For example, why did we reach certain fictions but not others?

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