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Dispensing the Indispensability Argument

Justin Yeary

The logician and naturalist Willard V.O. Quine makes the argument that mathematical objects

(sets, functions, groups, etc.) actually exist due to their indispensable utility to the natural sciences.

This amounts to nothing more than an argument from convenience, which is bad logic. The purpose of

this paper will be to disprove Quine's hypothesis, and make the case that although mathematical entities

do exist, the indispensability argument is not the way to go about proving this. This will be

accomplished in 4 sections. The first section shall be a brief overview of Quine's holistic naturalism, his

indispensability, and where it falls into the historical picture of the philosophy of mathematics. The

second part will consider Joseph Melia's paper “Weaseling Away the Indispensability Argument” and

use it as an argument against Quine's thesis. The third portion will be devoted to Alan Baker's “Are

There Genuine Mathematical Explanations of Physical Phenomena?”, which will serve as a counter-

point to my thesis. Finally, in the last section I will posit my own attack on the indispensability

hypothesis from a realist, rather than a nominalist, perspective.

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Background and History

Quine started to dabble the philosophy of mathematics in a period during which logical

positivism was the dominant school of thought. Logical positivism is a philosophy which attempts to

do away with metaphysics, focusing instead on the analysis of language through the use of rigorous

logic. Perhaps the most famous idea within the school of logical positivism would be that of the

verification principle. This principle states that all possible statements within ordinary language can be

partitioned into two categories: those whose truth values can be determined analytically through the use

of a logical decision procedure, and those whose truth values are indeterminate. To the positivist, only

the analytical truths, or so-called “cognitively meaningful” sentences, have any sort of relevance to

philosophy. A consequence of this radical view is that it completely does away with any branch of

philosophy whose truth is not empirically verifiable; this includes ethics and metaphysics as a whole.

To the positivists, ethical statements are nothing more than people conveying their emotional attitudes

towards a particular set of actions. Since it is not possible to objectively verify the truth or falsity of an

ethical statement, it follows that all ethical statements are junk.

Quine adamantly opposed the positivist stance. He viewed the verification principle as a self-

contradictory idea which attempted to create too concrete a separation of all linguistic sentences.

Although he appreciated the logical positivists focus on empiricism and high regard for science, Quine

felt they took their quest for scientific truth a little too far. He became a proponent of what is known as

holistic naturalism. In the article “Quine and the Web of Belief (2007)”, Michael Resnik defines

holistic naturalism as, “...the doctrine that no claim of theoretical science can be confirmed or refuted in

isolation, but only as part of a system of hypothesis” (Resnik, pg. 414). One implication of this holism

is the rejection of the positivist distinction between logical a priori truths and empirical a posteriori

ones. Science is a field of study which is constantly evolving, and it is perfectly sensible to change our

views on what constitutes truth in light of new experiences. This holism explains, for example, how

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science could suddenly shift from advocating a geocentric model of the universe to a heliocentric one.

Scientists such as Galileo discovered new empirical evidence, and as a result the underlying theory of

science changed as well. But what of other truths which have conventionally been considered

independent of experience and immune to revision, such as mathematics? Quine believes that there are

no such truths. As Resnik writes, “... no belief is immune to revision in the face of contrary experience”

(Ibid. 416). This is due to the fact that, “... experience bears upon bodies of beliefs and, insofar as it

may be said to bear upon individual beliefs of a system, it bears upon each of them to some extent”

(Ibid. 416). As science and technology evolve, so do our beliefs, and the experiences that arise as a

result of this evolution may change our perception of these seemingly unchangeable, Platonic ideals.

For example, the computer science revolution brought about by Alan Turing and his idea of a Turing

Machine completely changed the way modern mathematics works; whereas before the dominant trend

in mathematics had been a formal one, with great emphasis on rigorous proof, the advent of computers

has lead to an entirely new branch of mathematics known as experimental mathematics, which relies

less on traditional methods of formal proof, and more on statistical analysis and computations aided by

computer calculations. These changes in our perceptions of the world can lead to changing the way that

fields such as mathematics are treated.

As a corollary of Quines holism, he advocated what is known as the Indispensability Argument,

which is the main topic this paper is concerned with. Mathematics lies at the foundations of every field

of science, and thus by the holism which Quine advocated, in order to accept any scientific theory as a

source of truth, we must also accept all of it's baggage along with it as true. Therefore, due to

mathematics' indispensable role in science, we must accept the ontological status of mathematical

entities as true. As Resnik more eloquently puts it:

… mathematics is an indispensable component of natural science. Thus, by holism, whatever evidence we have for science is just as much evidence for the mathematical objects and the mathematical principles it presupposes as it is for the rest of its

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theoretical apparatus. Third, by naturalism, this mathematics is true, and the existence of mathematical objects is as well grounded as that of the other entities posited by science. (Ibid. 430)

This is Resnik's summary of Quine's argument, and it is the thesis which I shall be criticizing. I

view this as nothing more than an argument from convenience, which is a poor mode of

argument. Before espousing my objections, I will first consider Joseph Melia's critique of the

IA, which approaches the issue from a nominalist perspective. Although I disagree with Melia's

nominalism, I feel that he still makes many strong points which can be used to argue against

Quine.

Melia's Take on the IA.

In his paper “Weaseling Away the Indispensability Argument (July 2000)”, Melia does two

things: the first half of the paper argues towards constructing a nominalist theory of science and away

from a realist one by what he refers to as the “Trivial Strategy”. His strategy is to take a realist

scientific theory U, and construct from it a nominalist friendly theory T which has the same

implications in reality and theory as U. He then gives the reader a specific process by which to

construct such a T. He writes:

Given any unkosher theory U simply partition the predicates into two classes: those that are nominalistically acceptable and those that are not. Let theory T be those sentences that are logically entailed by U, yet whose vocabulary contains only nominalistically acceptable predicates. Since every nominalistically acceptable sentence logically entailed by U is logically entailed by T, it would seem as if T has all the kosher consequences of U and none of the unkosher ones, and thus can serve as the nominalist’s replacement theory. Since the reasoning here is entirely general, the nominalist appearsto have a winning strategy no matter what the Platonist theory U that scientists eventually settle on will be. (Melia 2000, 458)

Melia is basically saying that one could theoretically nominalize any scientific theory by only regarding

those sentences G within the theory U such that all G are concrete, thus satisfying the conditions of

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nominalistic scrutiny due to the inherently similar logical structures of U and T. How he proposes to

partition such a theory is left as a mystery; Melia doesn't offer any sort of criteria or guidelines one use

to partition any such theory. In this respect, Melia is extremely vague. I would conjecture that the

particular technique Melia proposes for his partition is to simply do away with any sentence that seems

to assert the existence of mathematical abstracta. He then goes on to write that although this T is indeed

nominalistic, it is less aesthetically appealing than it's counterpart U. But to the nominalist, aesthetics

are irrelevant; the issue at hand is ontology, and since T gets the job done in using the appropriate

vocabulary, it is a valid theory. However, for the mathematical realist, such an argument isn't quite

plausible.

One aspect of Melia's paper I feel necessitates commentary is his synonymous use of the words

Platonism and realism. Given that Melia is a nominalist, it is easy to see why he would lump the two

philosophical positions into the same category; anyone who argues that abstracta exist is essentially

arguing a bunch of metaphysical tomfoolery. However, I believe it is appropriate to bring the readers

attention to Melia's choice of nomenclature, as there is a distinct philosophical difference between

Platonism, which views mathematical objects as a priori objects which exist independent of space-

time, and realism, which merely asserts that mathematical entities exist (not necessarily in some

metaphysical Platonic realm).

Melia also considers what happens when one works in the opposite direction: going from a

nominalistic theory T to a Platonist theory T*. He comes to the conclusion, using the axioms of Peano

arithmetic to prove his point, that the vocabulary of T is too weak to express the powerful results of

Peano arithmetic which are used in describing natural phenomena. In Melia's words:

...the Platonist language enabled us to say something about the way the sums were that was not expressible in the nominalist language. By adding new predicates to our theory and new entities to our ontology, we were able to guarantee the existence of a certain kind of concrete entity whose existence we were unable to guarantee in the nominalist system. (Ibid. 466)

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Since nominalistic ontology does not allow quantification over abstracta, it loses much of the

expressive capabilities of Peano arithmetic, which quantifies over abstract predicates and variables in a

second-order formal system of logic. Therefore, statements such as “For all integers n, formula X” are

hard for the nominalist to express, since nominalism considers only concrete instances of phenomena,

and not abstract generalizations. It just so happens within science that those sorts of statements are

incredibly handy for expressing data. For example, the calculus of real variables, which plays an

integral role in many scientific theories, is based on the fundamental laws of Peano arithmetic applied

to functions, and to take a nominalist approach to science would require not only an entire rewriting of

basic arithmetic, but also a complete reconsideration of the concept of functions, which are the

foundations of calculus. It is in this sense that “the Platonist language enabled us to say something...

that was not expressible is in the nominalist language.” To get around this problem, Melia proposes that

it is reasonable for a nominalist to use T*, but with caution; in effect, it is perfectly consistent within

nominalist beliefs for one to assert the truth of T* with the additional criterion that the existence of sets

is denied. At first, this may seem to be a contradictory statement, but a deeper analysis reveals this is

consistent with nominalistic ontology. The mathematics of T* is used to explain and interpret physical

phenomena much like someone who wears glasses sees the world through their lens. Just because the

language of mathematics is convenient to explain phenomena does not mean that one must commit

themselves to a realist ontology. The basic idea is similar to how one uses a hypothetical situation to

prove a point: by definition, a hypothetical situation is not real. However, there is still utility in talking

about such a situation because of its explanatory power. In a similar fashion, mathematical statements

function the same way. You can use a certain kind of mathematics, such as set theory, because the

axioms and rules of set theory describe physical phenomena conveniently. However, it's not at all

necessary to commit ones self to the existence of sets; they're merely used as a fictional construct to

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prove a point. In his own words, “Joe is taking advantage of the mathematics in T* to communicate or

express his picture of what the world is really like.” (Ibid. 467)

One may find such an ontological stance to be a hypocritical one, but Melia assures us that this

is not at all the case. Just as one may weave a story in real life, only to modify certain parts of it later

one, Melia asserts that the same is possible of a scientific theory. He brilliantly summarizes the

dispensability of mathematics as follows, “The mathematics is the necessary scaffolding upon which

the bridge must be built. But once the bridge has been built, the scaffolding can be removed” (Ibid

469). Naturally, this begs the following question: If one is going to construct a theory only to do away

with certain parts of that theory, why even bother constructing it in the first place? Because

“Sometimes we have to” (Ibid 470.) The earlier example concerning the languages T and T* exemplify

this point: even though we could assert much of the world by using the nominalistic theory T, in

actuality it is not powerful enough to describe all the world due to its inability to express mathematics

as powerfully as T*. Thus, in order to capture a more perfect image of the world with the power of T*'s

mathematics while remaining loyal to a nominalist ontology, it is necessary to assert T* sans the

existence of sets. It is only through this slight modification that one can hope to analyze and understand

the world to any degree of utility. In summary, Melia writes, “It is surely more charitable to take

scientists to be weasels than hypocrites” (Ibid 470). Although this particular mode of argument

(modifying realist theories of science with “Sets don't exist”) is, indeed, a very weaselly form of

argument, it is nonetheless consistent and valid.

One objection raised by Hilary Putnam to Melia's weasel argument is that such a mode of

argument is completely irrational. Putnam writes:

It is like trying to maintain that God does not exist and angels do not exist while maintaining at the very same time that it is an objective fact that God has put an angel in charge of each star and the angels in charge of each of a pair of binary stars were always created at the same time! (Ibid 469-470.)

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Melia's retort to this objection is that this is not necessarily the case. As a counter-example, he uses a

guy named Joe who asserts a theory about stars, angels, and a God who creates everything. In this

theory, Joe asserts that:

In charge of each star is an angel, no two angels are in charge of the same star, and at the precise moment that each star is created the corresponding angel is also created. Moreover, the angels in charge of stars a and b were created at the very same time. (Ibid 470.)

Melia then proceeds to attach to this theory the claim that there are no angels, and asserts that this is not

contradictory, contrary to Putnam's concerns. Melia admits that his additional claim essentially amounts

to “Stars A and B were created at the very same time.” Although it's quite true that Joe is being long

winded, in no way does linguistic extravagance amount to any sort of irrationality or paradox.

Furthermore, not every case turns out to be as clean-cut as Joes. As was demonstrated earlier in the case

of theories T and T*, sometimes such modifications truly are necessary in order to capture all the

niceties one desires. The purpose of a scientific theory is to communicate knowledge about the world,

and if that involves using a theory which commits itself to the existence of abstracta, then so be it.

Simply tack on the denial of abstracta to any theory, and there is no contradiction of any sort. One can

stick with their clean nominalistic ontology, while simultaneous harnessing the expressive power of

Platonist mathematics.

The final portion of Melia's paper deals with the following objection: If it is scientifically valid

to accept the existence of concrete unobservable entities, such as gravity, why should mathematical

abstracta be treated any differently? Surely there is no physical, observable phenomena which one can

point to and say, “This here is gravity.” However, by admitting the existence of gravity in concreto, our

theory of physics is greatly simplified. From our assumptions on gravity, many other valid and useful

scientific hypothesis & conclusions can then be inferred. Numbers are even more fundamental than

gravity to all fields of science, and given that we admit the validity of the indispensability hypothesis

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applied to quarks, gravity and dark matter as valid, numbers should be treated no differently.

His retort essentially boils down to the fact that in scientific theories, numbers are often used as

an index for a specific formula or relation. Melia asks us to consider two theories, T1 and T2. T1

contains an infinite number of predicate relationships, such as “x is 3 meters from y”, “x is ½ meters

from y”, etc. T2 contains a single predicate of the form “x is r meters away from y” which quantifies r

over the real numbers. Both T1 and T2 have their strengths and weaknesses: T1 gives us specific

examples and exact quantities, but deals with an infinite number of relations. T2 gives us a single, clean

relationship but must quantify over an infinite number of values. It's understandable to see the

arguments in favor of either theory, but Melia finds this a misnomer, saying, “For, although T2 uses

fewer primitive predicates than T1, there is no reason to think that T2 actually postulates fewer

fundamental relations than T1” (Ibid. 473). In effect, T2 is just as complicated as T1 because although

T2 may appear “cleaner” at the surface, in actuality one must still instantiate the relationship an infinite

number of times, which makes it fundamentally the same as T1. He then writes:

Rather, the various numbers are used merely to index different distance relations, each real number corresponding to a different distance. But if numbers merely index the relations then no conclusions can be drawn about the nature of the relations themselves—in particular, no conclusions can be drawn as to whether or not the distance relations are fundamental. (Ibid. 473)

In effect, what Melia is saying is that unlike gravity or dark matter, which are unobservable

concretes accepted within the scientific community, numbers and mathematical objects do not

correlate to any physical entities; rather, they are fictional objects we create for the sake of

convenience and organization. Furthermore, in the example concerning T1 and T2, it ends up

being that neither theory is “simpler” or “cleaner” than the other, because T1 is essentially

nothing more than the sum of all particular instantiations of T2. Therefore, assuming the

existence of mathematical abstracta does nothing to clear up our scientific beliefs in the sense

that assuming the existence of gravity does.

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Melia ends his paper with the admission that it is possible for abstract mathematical

objects to exist, should the purely mathematical properties of said objects improve the overall

clarity and “neatness” of a scientific theory. In his own words, “And certainly, the defenders

need to do more than point to the fact that adding mathematics can make a theory more

attractive: they have to show that their theories are more attractive in the right kind of way”

(Ibid 474). By the “right kind of way,” Melia means that the purely mathematical properties of

abstracta must serve to clarify a particular scientific theory. This is where Alan Baker and his

paper, “Are There Genuine Mathematical Explanations of Physical Phenomena?” enters the

fray.

Baker's Response

Alan Baker's paper “Are There Genuine Mathematical Explanations of Physical Phenomena?

(2005)” jumps into the nominalism vs realism discussion concerning the indispensability argument as a

reaction to Joseph Melia's paper discussed above. At the end of Melia's paper, he makes the case that

the indispensability argument could be potentially valid should it offer the right kind of explanation. As

Hartry Fields wrote, Platonists must propose “one special kind of indispensability argument: one

involving indispensability for explanations” (Baker 2005, pg. 224). After offering a brief summary of

the ongoing dialogue (the details of which I shall omit, as I have covered them earlier in this paper),

Baker attempts to argue that this “special” kind of indispensability is indeed provable. Using the birth

cycles of cicadas as his case-in-point, Baker shows us an example of a concrete, observable

phenomena which is explained by the completely mathematical number-theoretic properties of prime

numbers. This phenomena, he posits, is one example which satisfies Melia's criterion: not only is the

phenomena in question explained in purely mathematical terms, but the mathematical properties have a

causal relationship with the phenomena; that is, the property of primeness is the genuine explanation

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for the frequency of the cicada's birth cycles.

Baker explains what exactly this phenomena is, and how prime numbers are related to it. He

writes:

Three species of cicada of the genus Magicicada share the same unusual life-cycle. In each species the nymphal stage remains in the soil for a lengthy period, then the adult cicada emerges after either 13 years or 17 years depending on the geographical area. Even more strikingly, this emergence is synchronized among all members of a cicada species in any given area. The adults all emerge within the same few days, they mate, die a few weeks later and then the cycle repeats itself. (Ibid. 230)

Upon examination, one notices that the life cycles of the cicadas both happen to be prime numbers.

This naturally begs the question of whether this is mere coincidence, or if there is some scientific

justification for the fact that the life cycles are primes. Baker argues the case that the life cycles of the

cicadas are prime is completely intentional. The cicadas evolved over time to have such a life span, and

having that number be a prime comes with many advantages which make sense within the context of

biological evolution.

Two explanations for the primeness of life cycles are currently accepted among biologists. The

first is that having a prime life cycle reduces the chances that the cicadas will encounter their natural

predators, who usually have life spans that are non-prime numbers. Since a prime numbers only factors

are 1 and itself, the chances that an animal with a composite (meaning not-prime) life cycle are greatly

reduced, based off the elementary number-theoretic concept of the least common multiple (LCM).

Baker writes,

“Assume that m and n are the life-cycle periods (in years) of two subspecies of cicada, Cm and Cn. If Cm and Cn intersect in a particular year, then the year of their next intersection is given by the LCM of m and n. In other words, the LCM is the number of years between successive intersections.” (Ibid. 231-32)

In terms of evolution, the cicadas want their life cycles to intersect with as low a frequency as possible.

Since any prime number m is coprime with all numbers n such that n < 2m (Ibid. 232), it follows that

by having this number be prime, the chances of a cicada being born while predators are roaming freely

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is greatly reduced, thus giving the insects a survival advantage. Also, since cicadas within the species

have life spans that are prime, it follows that their life cycles will be coprime with one another. This

offers another evolutionary advantage in the fact that the cicadas will not have to compete with one

another over resources such as food, territory, etc.

Having explained his example, Baker then proceeds to determine whether his cicada's fit

Melia's criteria of a genuinely mathematical explanation. He gives us 3 explicit criterion as to what

would constitute a mathematical explanation (although these criterion are not quite strict enough to

meet Melia's standard of a “genuine” explanation). They are that the problem in question must be one

external to the field of mathematics. Since Baker's example deals with the biology of insects, clearly

that condition is satisfied. Second, the phenomena in question must be in need of explanation. As Baker

mentions previously in his paper, the primeness of cicada life spans was very mysterious to biologists,

and it took a long time for any biologists to account for that fact, thus the second condition is satisfied.

Finally, the phenomena in question must have been discovered independently of the mathematics which

explains it. Baker notes that the primeness of magicicada life spans has been known for over 300 years,

which is older than the number theory from which the explanation is derived. Therefore, the third

criterion is satisfied, which signify Baker's cicadas as a mathematical explanation of a scientific

phenomena. However, the question still remains as to whether these insects constitute a genuine

mathematical explanation.

Before one attempts to answer that question, it is helpful to first answer the question of what it

is that makes a mathematical explanation “genuine” in nature. In Baker's words, this means that, “... the

mathematical component of the explanation is explanatory in its own right, rather than functioning as a

descriptive or calculational framework for the overall explanation” (Ibid. 234). In the case of the

cicadas, this is certainly true. For example, concepts such as distance can be represented by 3 place

relationships (X is R meters from Y) where numbers function as mere indexes for a more general idea

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Also, when differential equations, calculus, etc are used within science, their functions are purely

computational: the point of those mathematical endeavors is to produce a number which precisely

quantifies a more general scientific property, such as distance or pressure. The primeness of the cicada's

life span fits neither the indexing nor the computational role. The co-primeness of numbers and the

lemma concerning multiplicity of primes are all objects which offer a scientific explanation in purely

mathematical terms. 13 and 17 aren't just some arbitrary values chose to represent a particular

relationship, nor are they the result of some menial calculation to quantify a property; 13 and 17 are

specifically chosen for the fact that they are prime numbers. In short, the relationship between the

mathematical property of primeness and the life span of Magicicada is a causal one. It is this causality,

Baker argues, that makes his example a genuinely mathematical one.

Although Baker does an outstanding job at retorting Melia, he still fails to address some

fundamental ontological concerns regarding the indispensability argument. Melia also makes a few

strong points in his paper against the IA, but makes a fatal flaw in allowing there to be a possibility that

the indispensability argument is correct, which avoids addressing the fundamentally flawed logic in

Quine's argument. This is where I will step in with my own thoughts, attacking the indispensability

argument from a realist point of view while using some of the ideas formulated by Melia to support my

argument.

My Thoughts

In my view, the indispensability argument is poor ontology. I like to view it as a hypothesis

which smuggles abstracta into the realm of the real. The hypothesis boils down to an argument from

convenience: it is convenient for science to assume that mathematical objects exist as concrete entities

because mathematics explains so much about science, and the calculations which mathematics make

possible are all too important to science to simply discard. As a result, we must believe in the existence

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of abstract mathematical objects. Although I am sympathetic to the realist approaches to mathematical

ontology, I feel that Quine's hypothesis is a sneaky and logically unsound way of arguing for realism.

Mathematical entities do exist, but using Quine's indispensability argument to make the case for

realism is not the way to go about defending realism. One can attempt to use the IA in defense of

Platonism, but that does not change the fact that one is ontologically “cheating”.

To expand more upon what I mean when I say 'ontological cheating', consider the following

argument, which parallels the logical structure of the IA.

1) Morality is a necessity for a functioning society

2) Only through a belief in God can one ever possibly act ethically.

3) The non-existence of God would shatter the faith of the masses, and thus destroy any

semblance of morality

4) Therefore, for the sake of social stability, God must exist.

Clearly this mode of argument is absurd, regardless of ones beliefs on ethics and God. However, the

structure of the argument is not all that different from the indispensability argument. Replace God with

mathematics, morality & ethics with science, functioning society with the ability to know truth, and

you end up with Quine's indispensability argument. If it is absurd in the case of my hypothetical

example, why should anyone give Quine's thesis, which has the same logical structure, any

consideration?

Furthermore, the indispensability argument does nothing to directly prove the ontological status

of mathematical entities. The crux of the IA lies solely on the underlying assumption that science

amounts to the best way of knowing truth. I do agree that mathematics is indispensable to science, but

why is science given a privileged status which requires us to sweep mathematics into our ontology? To

the positivist, this is easy to see; empirical observations lie at the heart of positivist philosophy, and

thus one can understand how a positivist would come to such a conclusion. However, I do not identify

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as a positivist, and for me this isn't a very persuasive argument. Therefore, I ask the question “Why

science?” This should not be interpreted as me downplaying the importance of science, because I agree

100% that science is an extremely important body of knowledge that leads humanity to discover truths

it could never have possibly conceived of learning any other way. This does not mean, however, that

science is the best way we learn truth. When one is attempting to discover truths about the way our

world and our universe functions, science is unparalleled in it's efficiency and descriptive power. Thus I

would say for those sorts of truths, science is indeed the best way of learning truth. However, there are

other kinds of truths in the world which science cannot say anything about. For example, science

doesn't have much to say in regards to truths about my emotions and the way I feel, or truths

concerning the ideal system of politics, etc. There are many mediums through which truth can be

conveyed, and science just happens to be one of those. Truth can be found in art, for example, just as

much as it can be found within science. The truth one realizes from admiring a Picasso painting is of a

different nature than the truth one realizes from studying physics, but this doesn't mean artistic truth is

any less valid than mathematical or scientific truth; it is simply of a different breed. An implicit

assumption of the indispensability argument is that science is elevated above all other fields of study,

but there is no logical reason to believe this.

A perplexing logical consequence of accepting the indispensability argument as true is that it

implies mathematical objects can phase in and out of existence. For example, the haversine function,

which was at one point used widely within the nautical sciences as a tool for making trigonometric

calculations, has gone nearly extinct due to the advent of computers. Under the indispensability

argument, that would imply that the haversine function no longer exists, since it is no longer useful or

necessary for science. Yet this seems to me an absurd proposition, because even though the haversine

might not receive much use, the mathematics behind it are still perfectly legitimate and remain valid to

this day; there just aren't many practical applications for it. Similarly, some of the most modern,

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cutting-edge mathematical developments have yet to find a practical application. This isn't to say that

one day a use for these advanced mathematics will not be found, but what it does imply is that these

ideas will not be real until science needs them for it's own purposes. This is also a perplexing quandary,

because even though the mathematics behind these developments is completely valid and

mathematicians can explicitly formulate such ideas in the context of a formal system, the

indispensability argument tells us these things do not exist, since science doesn't make use of them.

However, once science finds a use, they will become as real as the computer I am typing on. I find this

idea that mathematical objects can phase in and out of existence to be an utterly absurd one, and an

unintended logical consequence of mathematics.

Although I disagree with Melia's nominalist sentiments, I cannot help but agree with some of

his critiques of the indispensability argument. In particular, I find his argument in favor of saying

“Theory T*, but without the existence of sets” to be very persuasive. Much like an artist uses their

imagination as an inspiration for their art, the scientist can use the “fiction” of mathematics in a similar

fashion to develop their particular theory. Just because one uses a fiction as a stepping stone in

explaining their point, does not mean that this fiction suddenly becomes a reality, no matter how

indispensable this imaginary object is to the story. Therefore, there isn't really a need for scientists to

accept mathematical abstracta as concrete, because these concepts harness the same explanatory power

regardless of their ontological status.

Assuming that the indispensability argument is true, this means that sets actually exist, and from

the existence of sets, one can derive the concrete existence of functions, relations, etc in the usual

fashion. But this is clearly an absurd view! Mathematical objects contain many properties which are too

ideal to correlate to anything in reality. For example, one can consider a sphere in Euclidean 3-space.

Such a sphere has many nice and idealistic properties, yet in real life, there is not a single object that

translates perfectly to a sphere. It is true that scientists have designed sphere-like objects which differ

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from an ideal sphere on a near-infinitesimal scale (the margin of error is merely a few atoms), but

nowhere in nature can we find a perfectly smooth surface like a sphere. Similarly, one can consider a

“nice” function, like a polynomial spanning over the reals. Although these functions are differentiable

everywhere, smooth, continuous, integrable, analytic, etc. one can never find an actual object, data

trend, or phenomena of any sort which is perfectly smooth, always continuous, etc. Once again, the real

world offers us some incredibly close approximations, but these ideal properties can never be achieved

in physical reality. Thus, I fail to understand how one can argue that these perfect polynomials and

spheres can possibly exist, while not being able to point to a single object or phenomena in the real

world that correlates to such an ideal mathematical object.

In Melia's paper, the gravity objection is raised; scientists generally agree that unobservable

phenomena, such as gravity, are as real as the chair I am sitting in. Since we accept that these

phenomena are concrete, why not accept the existence of mathematical objects as well? Although it's

true that we cannot directly observe things like gravity, there are experiments that can be performed

which agree with the existence of such entities. For example, if I throw a ball up into the air, gravity

tells me that ball will eventually fall down, and by throwing a ball into the air I will see that my

prediction is true. Although we cannot point to any single object and say “Look, this is gravity!”, we

can perform all sorts of experiments which produce empirical data that complies with the assumptions

of our theory. The same cannot be said of mathematical abstracta. I cannot conceive of any sort of

empirical experiment one could perform to say, “Yes, the data clearly shows that this particular

function can be expressed as a power series over an infinite radius of convergence!!” I do not even

believe such an experiment to exist! Therefore, the ontological status of mathematics fails when

examined under the same scientific scrutiny which we subject other unobservable, concrete phenomena

to.

Two objections may be made to the above argument concerning experiment, both of which I

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wish to address. The first is concerning my claim that there is no experiment to show us a mathematical

truth, such as 1+1=2. One may ask in response, “Well then, what do you make of mathematical proof?

Aren't proofs the mathematical equivalent of a scientific experiment?” I hesitate to use the words

“mathematical equivalent” to compare proof to experiment; I feel that the terminology “mathematical

analog” is far more appropriate. Much like a well-designed experiment leads us to discover new

scientific truths, a well-designed mathematical proof leads us to discover new mathematical truths. But

where experiment and proof differ is in their origins. In accordance with the scientific method,

experiments must be grounded in empirical phenomena and based upon observable evidence. But this

is not true of a mathematical proof. Proofs boil down to logical rules of inference and a set of axioms

which are used to construct objects and theorems within a particular formal system. However, these

axioms aren't observable, nor do they correlate to any sort of tangible object. I challenge anyone who

opposes this claim to explain to me how one could possibly hope to observe the phenomena of the

axiom of choice, or any other axiom of ZMF set theory in concreto. It is this divorce from empirical

phenomena which constitutes the difference between a scientific experiment and a mathematical proof.

The second conclusion which can be made to my experiment argument involves the field of

experimental mathematics. One could raise the objection that, thanks to the advent of modern

computers and calculating tools, mathematicians can now perform experiments with numbers and sets

on their computers. A computer that can add numbers with a trillion digits can certainly offer us some

deep insight into unproved theorems and conjectures, and this should qualify as an experiment. While I

will admit that these computing systems are impressive, and that they do give mathematicians a

powerful new tool with which to make discoveries, they don't go nearly as far as a proof does in

conveying to us deep mathematical properties. It is true that computers can solve some theorems which

humans have struggled with for centuries; the computer assisted proof of the four-color theorem shows

us that in instances where a proof defends on a finite and discrete cases, computers can greatly aid the

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mathematician in making mathematical discoveries. But it was the insight of some mathematician who

intuitively knew the theorem to be true that caused the development of a computer program to solve the

four color theorem. Also, for theorems whose scope is the infinite such as the Goldbach conjecture,

computers cannot ever hope to prove such a proposition by finite cases, since there are an infinite

number of primes. It is true that the computers can perform amazing arithmetical computations with

extraordinarily large numbers, and it's also entirely possible that in performing these computations, the

computer may find an even number which cannot be represented as the sum of two primes. But never

can we hope for a computer-created proof which settles the case in the affirmative; only a

mathematician can do such a thing, because computers lack the creative insight and intuition which

mathematicians regularly make use of in constructing their proofs. Computers might do a fantastic job

at performing algorithms and arithmetic, but at the end of the day, no machine can replace the creativity

and gut feelings of a human mathematician. Thus, we cannot hope that experimental mathematics will

ever reveal to us insights which are analogous to those that scientists gain through experimental

observation. Furthermore, experimental mathematics suffers from the same criticism I made in the

paragraph above; it still fails to tie itself down to observable, empirical phenomena based in reality, and

thus to call it an experiment is a bit of a stretch.

In summary, the indispensability argument, though it might sound plausible at first, is really

nothing more than an exercise in poor logic that stems from a misunderstanding of what exactly makes

science such a good source of knowledge. The structure of the argument amounts to nothing more than

an argument from convenience. If arguing from convenience were a logically valid strategy, then it

would lead to all sorts of absurdities; thus, why should Quine's hypothesis be privileged in making use

of it? Furthermore, mathematics and science operate on fundamentally different levels when it comes to

ascertaining knowledge. Science tells us about the world around us by observing empirical phenomena,

collecting data, connecting the dots, and drawing conclusions from data which is grounded in reality.

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Mathematics, on the other hand, is based on using logical rules of inference to combine axiomatic

statements which allow us to communicate ideas which come from inside our minds. I have stated

previously in this paper that although I think the indispensability argument is invalid, I am still a realist.

This necessarily begs the question that if I don't take fancy to the IA, then what sort of realism

do I advocate? The best way to describe my brand of realism is as a special brand of intuitionism,

although I am hesitant to use that word due to it's associations with radical matehmatical concepts, such

as the denial of the law of excluded middle and of the varying degrees of infinity. Rather, I use the term

intuitionism because I sympathize with the intuitionistic notion that mathematics is an activity of

creation, a by-product of the creative endeavors of human intellect. I believe mathematical objects exist

within the confines of our minds. This doesn't imply that mathematical objects reside in some

transcendental a priori metaphysical realm, but rather that mathematical objects are a construction of

the creative capacities of the mind, and it is in this sense that they are real. A good metaphor to explain

exactly what I mean by this is qualia. Qualia are those properties of experience we all know to be true,

but are fundamentally subjective. For example, consider the property of redness. We all know that there

is a color such as red, but when we go to describe what exactly this color looks like, we come to a loss

for words. That is because redness is a subjective property of the mind; although we agree that the same

object is red, it is entirely possible that my red might look completely different from your red.

Mathematical objects are much the same, though they lack the degree of subjectivity associated with

qualia. Mathematical objects are fundamentally subjective, but this subjectivity arises not in the final

product of the object itself, but rather in the process of creations which an individual uses to construct

the object. They exist within the mind, and I believe that the existence of sets is something so obvious

and intuitive that we make use of them without even knowing it from day one. Even small children

know how to abstractly categorize concrete objects into sets, based on shared properties among the

objects. But to insist that a set is some sort of concrete physical phenomena on par with gravity and

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dark matter, as Quine suggests, is a ludicrous concept; the concept of set it something that is inside all

humans, and it is for that reason we cannot point to a single thing and say “Look, this is a set!”

Although I do take mathematical objects to be a special form of qualia, what differentiates

mathematics from other kinds of qualia is that it is completely possible to express mathematical objects

using an objective, formalistic language that is governed by logical rules of inferences. I have a great

appreciation for formal rigor and logic as a mathematician, but I do not believe that this rigor is the sole

purpose or object of mathematics. Rather, the real meaningful mathematical content are those ideas that

brew inside the head of the mathematician. Formalist interpretations simply provide the mathematician

with a means by which they can universalize and objectively express those essentially subjective

mental constructs which lurk within the mind of a mathematician. It is in this sense I believe

mathematical objects to be quite real, and the indispensability argument is a poor approach to proving

this brand of realism. Not only is it illogical, but the IA misses the true point of mathematics.

Mathematics is not there to simply clarify the scientists ideas. That is one possible use of mathematics,

but it is not the only one. Mathematics is created for it's own sake, it's own beauty, and it's own

purpose. Mathematics is a tool to express those beautiful ideas and objects which lurk deep within the

human mind.

Regardless of ones thoughts about my particular interpretation of mathematical realism, I've

laid out a few strong arguments against Quine's indispensability argument, and have also refuted some

common objections to such critiques. My purpose was not to convince the reader of the validity of my

own views, but rather to show the reader that Quine's approach to mathematical ontology is a

fundamentally flawed one, while still taking a realist approach to mathematical ontology, as most

critiques of Quine are rooted in nominalism. I sincerely hope that anyone who reads this paper walks

away thinking the same, or is at least willing to reconsider their views in light of this criticism.