The Indispensability Argument
The “Indispensability Argument” is an attempt to justify the Platonist belief that mathematical entities are actually real things that exist independently of any human concepts. It was Willard Quine who started the indispensability argument ball rolling. Since then various people, such as Hilary Putnam, Michael Resnik, Alan Baker and Mark Colyvan have continued to keep the ball rolling. The details of their arguments are given below. Although the argument comes in a number of different guises, essentially each one is some sort of variation on this (see also the section Ontological Commitment later on this page):
- Mathematics is indispensable (is essential) for scientific theories,
- The mathematical entities (such as numbers) referred to by the mathematical language must ‘exist’ in the same way that other things (such as electrons) referred to by the theory must ‘exist’.
On the face of it, this is ridiculous - there actually is no logical inference from assertion (1) to assertion (2). For example, while mathematics includes the notion of infinite quantities, scientific theories generally don’t claim the existence of infinite quantities, and for a theory to be considered scientific, it should be testable. (Footnote: See Karl Popper on testability and scientific theories. ) And there aren’t any testable theories that postulate an infinite quantity that can be physically tested.
But the fundamental problem with the indispensability idea is that it leads absolutely nowhere. It doesn’t lead to any new ideas, it doesn’t lead to any insightful theories. Its only purpose is to try to provide a justification for a belief in the real non-physical existence of things that somehow correspond to mathematical concepts. And in the same way as the unfounded pleadings for the existence of a god don’t result in any insight into the world that we live in, the indispensability idea doesn’t result in any advancement of either mathematics or science - it is an utterly useless argument - and thus, ironically, dispensable.
Although there is no real substance to the argument, and although it provides no useful insight, you will find different versions of the argument try to dress it up in ways that give a superficial veneer of acceptability. But it is easy to show that the basic argument is absurd. Every scientific theory is written/spoken/digitized in some language, and so we can consider a similar argument:
- A language (besides a purely mathematical language) is necessary for all scientific theories,
- The non-physical entities referred to by that language must ‘exist’, in the same way that other things (such as electrons) referred to by the theory must ‘exist’.
Yes, an equally absurd argument, with no logical inference from (1) to (2). A purely mathematical language is not sufficient by itself to formulate a scientific theory; the theory must use a different language to refer to real physical things, and the theory must correlate these physical things to the mathematics of the theory. But here’s the key point. Scientific theories rely on various idealizations which refer to imaginary ideal things. Physics may refer to infinitely long wires, perfectly uniform wires and ideal magnetic, electrical and gravitational fields that are infinitely smooth, infinitely divisible (regardless of any actual sizes) and perfectly uniform, and so on. Chemists may refer to reactions taking place at an ideal precise fixed temperature. They may use an idealization of a gas as consisting of all molecules having precisely the same temperature. Biology may use idealizations such as perfectly uniform populations. Economics may use idealizations such as perfectly rational consumers. Sociology may use idealizations such as perfectly average families.
Nobody believes that these idealizations actually ‘exist’. But if mathematical entities must ‘exist’ because a mathematical language is indispensable for a scientific theory, then it follows that such non-existing entities such as mentioned above must also ‘exist’, since they are referred to by a language that is indispensable for the scientific theory. (Footnote: Of course, there can be more than one language in which the theory can be expressed - but at least one such language is essential. ) This, of course, is absurd. And the notion that ideal mathematical entities also ‘exist’ is equally absurd.
Proponents of the indispensability argument often refer to a mathematical statement such as “3 is prime” and claim that since that statement is “true” (by which they actually mean that it is provable from mathematical axioms), then “3” must exist as an actual independently existing abstract thing. (Footnote: For example, in the book Platonism and anti-Platonism in mathematics, Mark Balaguer claims to be agnostic on the question of Platonism, but the essence of his writings are that since Platonism cannot prove the existence of mathematical things, and since non-Platonists cannot prove the non-existence of mathematical things, then there is nothing to choose between them. There is a striking similarity here to the theist’s argument that since one cannot prove the non-existence of a non-physical god, the theist’s view is as valid as a the non-theist’s view. See Mark Balaguer and Platonism. ) The absurdity of that claim should be obvious. Take, for example, the notion of the color red. Red is the color we see when our eyes receive light which has a wavelength between roughly 625–740 nm. Saying, “3 is prime” does not imply that there is an actual thing that is “3” any more than saying, “Red is a color whose wavelength is at the long end of the visible spectrum” implies that there is an actual independently existing thing that is “red”.
Problems with Beliefs About Mathematical Objects
Some philosophers talk about things that are referred to by mathematical statements as “mathematical objects”, but why call them “objects” at all? This is a sneaky way of implicitly asserting that mathematical entities are similar to relatively large physical things - we normally reserve the word “object” for a relatively large physical thing. We wouldn’t even call an electron an “object”, so why refer to mathematical concepts as “objects”?
Of course, there are mathematical statements that appear to correspond - approximately - to real physical situations, such as an ideal sphere approximating the earth. Or we can see that there is a common emergent property shared by five apples and five cats, and we call that common emergent property the number “five”. But that does not imply that there is some independent real thing that is “five”, any more than there is an independent real thing that is “living”, even though we can identify that certain collections of molecules (such as humans) can have the emergent property of “living”.
The web-page The Internet Encyclopedia of Philosophy - The Indispensability Argument claims that:
Mathematical objects are not the kinds of things that we can see or touch, or smell, taste or hear. If we can not learn about mathematical objects by using our senses, a serious worry arises about how we can justify our mathematical beliefs. The question of how we can justify our beliefs about mathematical objects has long puzzled philosophers.
This is nonsense. Why should there be any such concern? Mathematics arose because people observed that in the real physical world, the emergent numerical properties of collections of physical objects obeyed certain rules (that is, they observed inductively that the numerical properties obeyed certain rules), such as observing that if you have two sheep and you add five sheep, then you get seven sheep, and if you have five stones, and you add two stones, then you have seven stones. And so on. The fact is that the origin of mathematics came directly from observation of the real physical world.
The problems only arose when certain mathematicians decided to ignore any connection between mathematics and the real physical world, and decided to create theories that they called mathematics but which had no derivation from the real world. Much of the blame for this can be seen to have arisen from various notions attached to set theories. The web-page The Internet Encyclopedia of Philosophy - The Indispensability Argument goes on to say:
For simplicity, to consider our question of how we can justify our beliefs about mathematical objects, we will only consider sets. Set theory is generally considered the most fundamental mathematical theory. All statements of number theory, including those concerning real numbers, can be written in the language of set theory…
This again is nonsense. All of the mathematics that deals with the real world can be done, and in fact is almost always done, without reference to set theories at all. In terms of the history of mathematics, set theories are latecomers. The fundamental foundations for all of the mathematics that is used in today’s engineering and science was developed without any use of set theories at all. What the above conveniently omits is that each set theory (and there are several competing theories) introduces many notions that are not used at all by today’s engineers or scientists. (Footnote:
‘The necessary use of higher set theory in mathematics of the finite has yet to be established. Furthermore, a case can be made that higher set theory is dispensable in scientifically applicable mathematics …’,
Solomon Feferman: ‘Infinity in Mathematics: Is Cantor Necessary?’ Philosophical Topics 17.2 (1989): 23-45, also in the book “In the light of logic”, Oxford University Press (1998). ) This means that the idea that you must begin by considering set theories if you want to examine the relationship between mathematics and science is inherently ridiculous. It is a perfect example of self-serving nonsense - it is saying that set-theoretical entities must exist because, if we force a scientific theory to use set-theoretical mathematics, then we can claim that the set-theoretical entities must exist in the same way as other entities referred to by the theory.
To add to the absurdity of using set theory in an attempt to examine the relationship between mathematics and science, the web-page The Internet Encyclopedia of Philosophy - The Indispensability Argument goes on to say:
…given that we lack sense experience of sets, … how we can justify our beliefs about sets and set theory?
Well, of course, since set theory prides itself on being a theory that has no need of any foundation that is based on the real world, that is bound to be a problem…
For more on Platonism see Platonism, Mark Balaguer and Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.
Quine’s Indispensability Argument
Although it was Quine who initiated the indispensability argument, he didn’t write a specific article on it, but referred to it in various articles. (Footnote:
“Designation and Existence” In Feigl and Sellars, “Readings in Philosophical Analysis”, Appleton-Century-Crofts, Inc., New York (1940).
“Posits and Reality” In The Ways of Paradox, Cambridge: Harvard University Press (1976).
“Ontological Relativity and Other Essays”, New York: Columbia University Press (1969).
“From a Logical Point of View”, Cambridge: Harvard University Press (1980).
“Theories and Things”, Cambridge: Harvard University Press (1981).
“Philosophy of Logic”, Cambridge: Harvard University Press (1986).
“Reply to Charles Parsons” In Hahn and Schilpp, eds., “The Philosophy of W.V. Quine”, La Salle: Open Court (1986).
“From Stimulus to Science”, Cambridge: Harvard University Press (1995). ) A typical presentation of what his argument amounts to is this:
- We should accept the theory which best accounts for our sense experience.
- If we accept a theory, then we must believe that the things it refers to “exist”.
- The scientific theories which best describe our sense experiences use mathematics which refers to mathematical things.
- Therefore we should believe that mathematical things “exist”.
Yes, it is a ridiculous argument. But for some reason, Quine is considered by many philosophers to have been a philosopher of some substance. I don’t know why, I find that he is often quite illogical, see for example Quine’s Paradox.
Quine’s argument fails in its second premise; Quine argues that if you believe that a theory is correct, then you must believe in every object it posits. But this is simply playing games with the concept of belief. For example, I can accept Einstein’s theory of general relativity as being currently the best theory that accounts for various real world physical phenomena, but that does not infer that I consider that it is a complete description of the physical world. It isn’t. It’s a good approximation to certain physical situations, but it is not, by any standard, an exact description of reality. We can accept that scientific theory is useful, without presuming that any scientific theory is precisely “true”. That being the case, then why should I, or anyone else for that matter, want to consider that every conceptual entity utilized by the theory is in some sense a real thing, existing independently of any human theory? That’s absurd.
In science, idealizations are made simply to facilitate the construction of a theory. And so we can use the theory to generate useful results, while recognizing that some of the things that it refers to are idealizations. This is called being instrumentalist about some things that the theory refers to - they are tools used to produce a result. In the same way we can consider that mathematical entities such as numbers are simply tools that we use to produce a result. It could be argued that, while there may be better theories in the future, it is unlikely that the fundamental mathematics used by scientific theories is unlikely to change. But that tells us nothing, other than certain tools are particularly useful for certain tasks.
Putnam’s Success Argument
Hilary Putnam gives a slightly different argument: (Footnote:
“Mathematics and the Existence of Abstract Entities” Philosophical Studies 7 (1956): 81-88.
“Mathematics, Matter, and Method: Philosophical Papers”, Vol. I. Cambridge: Cambridge University Press (1975).
“Philosophy of Mathematics: Why Nothing Works” In Putnam, Words and Life. Cambridge: Harvard University Press (1994). )
- Mathematics is successful in its use in scientific theories
- There must be a reason for the success of mathematics in scientific theories
- No positions other than realism in mathematics provide a reason.
- Therefore, realism in mathematics must be correct.
The flaw in the argument is obvious, since the reason for the success of mathematics in science is obvious - because, as noted above, mathematics originated directly from observation of emergent properties of the real physical world. So it’s no surprise that it is successful as a tool to describe the physical world, and we don’t need the additional baggage that Putnam postulates.
Resnik’s Pragmatic Argument
Michael Resnik’s argument is essentially: (Footnote:
Resnik, Michael D,
“A Naturalized Epistemology for a Platonist Mathematical Ontology.” In Sal Restivo, et. al., eds., Math Worlds: Philosophical and Social Studies of Mathematics and Mathematics Education, Albany: SUNY Press (1993).
“Scientific vs. Mathematical Realism: The Indispensability Argument”, Philosophia Mathematica (3) 3: 166-174 (1995).
“Mathematics as a Science of Patterns”, Oxford: Oxford University Press (1997). )
- In stating its laws and conducting its derivations, science assumes the existence of many mathematical objects and the truth of much mathematics.
- These assumptions are indispensable to the pursuit of science; moreover, many of the important conclusions drawn from and within science could not be drawn without taking mathematical claims to be true.
- So, we are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true.
- We are justified in using science to explain and predict.
- The only way we know of using science thus involves drawing conclusions from and within it.
So we are justified in taking mathematics to be true.
Resnik’s claims are flawed from the outset. The initial claim is simply wrong. Science uses hypothetical mathematical entities, there is no necessity to assume any real independent existence of those entities. As for assuming the “truth” of much mathematics, what this is actually saying is that there is an assumption that the various mathematical statements are provable from an initial set of mathematical axioms and rules - there is no need for any notion of mathematical “truth”, only mathematical provability. Since the rest of the argument relies on this first premise, the remainder can be summarily dismissed.
Baker & Colyvan’s Explanatory Argument
Alan Baker (Footnote:
“Mathematics, indispensability and scientific progress” Erkenntnis 55.1 (2001): 85-116.
“The Indispensability Argument and Multiple Foundations for Mathematics” The Philosophical Quarterly 53.210 (2003): 49-67.
“Are there Genuine Mathematical Explanations of Physical Phenomena?” Mind 114 (2005): 223-238.
“Experimental mathematics” Erkenntnis 68.3 (2008): 331-344.
Mathematical explanation in science” The British Journal for the Philosophy of Science 60.3 (2009): 611-633.
“Science-driven mathematical explanation” Mind 121.482 (2012): 243-267 ) and Mark Colyvan (Footnote: Colyvan, Mark,
“The Indispensability of Mathematics”, Oxford University Press (2001).
“Mathematics and Aesthetic Considerations in Science” Mind 11 (2002): 69-78.
“Mathematical Recreation versus Mathematical Knowledge” In Leng, Paseau and Potter, eds, “Mathematical Knowledge”, Oxford University Press (2007): 109-122. ) have defended an explanatory argument, where the key point supposedly revolves around the fact that mathematics helps to explain physical phenomena, rather than talking about a scientific theory:
- There are genuinely mathematical explanations of empirical phenomena.
- We ought to be committed to the theoretical posits in such explanations.
- We ought to be committed to the entities postulated by the mathematics in question.
This, of course, is simply wishful hogwash, and is simply word-play on words such as “explanation”. A scientific theory can be a predictive theory, and predict outcomes of physical situations but if it does not help to explain things that is incidental. For example, there are many disputes over what quantum mechanics “explains”, even though its predictive power is immense. Neither Baker nor Colyvan add anything of substance to already mentioned vacuous indispensability arguments.
Some Platonists use the term “ontological commitment ” as the basis for their version of an indispensability argument. Trying to pin down what Platonists actually mean by the term “ontological commitment ” is rather like trying to find the pot of gold at the end of a rainbow. There are definitions that refer to ontological commitment as simply referring to some things according to a different definition than the conventional definition, the classic example being legal jargon where a “person” may actually mean an actual person or it can also mean an organization. But that is not what Platonists want it to mean; they want the “ontological commitment ” of a language to mean something along the lines of: if the language can be interpreted as referring to objects then it is necessarily implying that those objects exist, and so if you use that language you are admitting that those objects actually exist.
But of course, it would be absurdly circular to use that definition to try to prove that those objects exist. So you will find alternative definitions that try to evade that circularity. But whatever way the essence of that circularity is squeezed into an ill-fitting bottle by an attempt at a definition, the salient fact is that expressions of a language do not of themselves refer to objects. An expression of a language is merely a sequence of symbols. It is only by human convention that we agree that various words are to be associated with physical objects, where an initial ostensive physical gesture is required (an initial ostensively agreed isomorphism between the physical sequences of symbols of the language and other physical objects). In this way humans can agree that an expression of a language refers to a physical object. The language itself cannot create that association, the association is created by physical means outside of the language - if one were to show an alien, with no knowledge of human culture, sentences of human language that refer to physical objects, the alien cannot deduce what the expressions refer to.
The above applies for physical objects. On the other hand, if humans do not agree that certain words must be associated with imaginary non-physical objects that Platonists claim “exist” independently of any language, then no amount of pleading by a Platonist is going to prove such an association, and nor can any definition.
It might be the case that if one were to show the alien a selection of purely mathematical expressions, he might be able to deduce what the equivalent expressions are in his own mathematical language. This would not demonstrate that the human and the alien have both mystically gained the same knowledge of some type of non-physical objects, but it would suggest that the mathematics of both arose directly from empirical observations of the physical world. And the alien might be able to deduce what certain expressions of our language that describe things at the atomic and sub-atomic scales mean. But why would that imply that both humans and the alien are describing some sort of non-physical entities? Surely it would simply mean that at a fundamental level, the descriptions of real physical things are going to be similar simply because there is less room for ambiguity in the descriptions?
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