Ontology and mathematical entities: what are numbers?
Updated: Nov 29, 2021
When we talk about what exists, there are generally two strategies to answer: (a) the Quinean answer, which affirms that everything exists; (b) and the Carnapian answer, which affirms that there is what determines the domain of a given linguistic framework.
Obviously there are common sense responses that seek to affirm that there is what we see, hear, feel, and that this constitutes, objectively or subjectively, what exists for us. However, there is difficulty for common sense views as to justify the existence of entities such as numbers, since they are objective entities and not space-time.
In this short article we will see (1) what abstract entities are and David Lewis's strategy to characterize them; (2) what are mathematical entities, following Frege and Quine; (3) the famous argument of mathematical requisiteness, in which the existence of mathematical entities is defended; (4) and an epistemological problem to the position in favor of the existence of numbers.
1. What are abstract entities?
Abstract entities are entities that belong to a third realm, if they exist that meet a series of conditions or pathways described by David Lewis (1986): (a) the way of negation, (b) the way of example, (c) the way of inflation and (d) the way of abstraction.
The way of negation is the most obvious, as it describes the properties of abstract entities by denying those properties that make concrete entities what they are. Thus, following Rosen (2001), abstract entities would lack space-temporality as well as being causally inefficient. Now, there are entities that we consider abstract that seem to have a certain relationship with space-temporality, even though they do not exist in such, as is the case with artifactual entities: entities that begin to exist in an abstract way once created. This is the case with chess, novels or musical pieces. Likewise, these kinds of entities seem not to be causally inert, but rather come from or cause some kind of causality, at least in us. For example, the fact of reading 'War and Peace' or 'Capital', as thoughts can cause changes in the world that would not occur in a purely physicalist world, since they produce in us the felling of trees, revolutions, or forms of understand and conceptualize reality.
The way of example identifies the abstract entities based on the epistemology that causes the concrete / abstract distinction; For example, the causal theory of knowledge is fully functional for concrete entities, but it is not valid for abstract entities, and once some elements of each have been identified, by similarity we understand which entities are abstract and which are concrete.
The way of inflation looks for other pairs of distinctions to continue in the comparisons between entities, such as the distinction between particular and universal.
Finally, the way of abstraction is a classic way of identifying abstract entities because, since classical Greece, the philosophy of the anima understood that we have the ability to abstract concepts from the physical and concrete basis.
Now this does not justify or clarify why we should believe that mathematical entities are abstract even though it seems to us that this must be their nature. Two points will now be made in favor of (a) that they are abstract entities (b) and that they exist, since they are indispensable for our scientific practice.
2. The nature of mathematical entities
Following Horsten (2019), there are several ways of conceiving what type of entities are mathematical entities. Although it is a very stereotyped division, we can divide the positions regarding abstract entities into two groups: (a) the Platonists, who are those philosophers who affirm the existence of abstract objects (b) and the nominalists, those who deny their existence. Although Plato is indeed a Platonist, not all Platonists follow Plato's ideas, just as contemporary nominalists do not follow medieval nominalists. Later we will see Platonism and how Quine's indispensability argument to Platonism over nominalism:
If one is to judge between nominalism and realism on that basis, it is clear that the merits of nominalism diminish. The reason for admitting numbers as objects is precisely their efficacy in the organization and explicit accommodation of the sciences. (Quine, 2001: 299).
But, first you have to understand what they are and in what plane of reality they can (if any) be.
We know from the ontological commitment thesis that when an x can be a variable, it is an object. In chess there are rules such as "The king moves through squares one by one", but affirming this requires that for something to be subject to a rule, since that which affects the rule exists. That a king moves through squares implies the existence of a king, because if he did not exist, he would not have such a rule to follow. Be careful that, here when we speak of non-existence we speak of material and mental non-existence, since we have ideas and imaginary entities that have certain properties, such as that "Don Quixote's horse is called Rocinante", as opposed to impossible objects or Meinongnians.
Therefore, affirming that the king moves through squares one by one implies that there is at least one king, either from a Carnapian linguistic framework (if we assume a deflationary conception of ontology), or as something that really exists ( whether we assume a naturalistic or non-deflationary conception). In this sense, we can intuit that when we say one plus two equals three, we are implying that there is a one, a two and a three, ... that there are numbers. It would be a bit strange to say that "the sum of 1 plus 2 equals 3" but that the addends did not really exist, in the way that McX assumes Platonic entities as sets or attributes without an analysis of language.
There are at least three orientations to answer the question that a mathematical entity is: (i) mathematical entities are mental entities or constructs or are things that are in our mind, as innate ideas; (ii) they are physical entities, somehow space-time, and they are found in the world, they are entities of which we have experience as we have experience of a table; (iii) or they are abstract entities, which are neither mental nor physical.
Numbers appear to be abstract entities, and by abstract entity philosophers deal with objects that do not occupy a spatial and / or temporal location, and also that do not have causal relationships with other things. In fact, numbers are one of the first candidates to be abstract objects. However, even though we affirm that the number 5 is an abstract entity, this does not mean that it is a universal, they are particular because they appear as values of variables.
A first defense of mathematical entities was made by Gottlob Frege (1848-1925) at the root of his insistence on the objectivity and a prioricity of mathematical truths, which implies that numbers are neither material entities nor ideas of the mind. . The problem with the hypothesis that numbers are physical entities is that they had left it, or had properties of material entities, the laws of arithmetic and geometry would have the status of empirical generalizations, when since ancient Greece we have the intuition that mathematical knowledge is special in the sense that we do not need the world to know the veracity of our mathematical hypotheses. For example, "it is raining outside right now" has a different way of presenting itself to me than "two plus two equals four", because while the first statement seems to be able to be true or false, the second appears to be necessarily true.
On the contrary, if the numbers were ideas of our mind, the same problem would appear as well as problems of subjectivity as if when I say '5' and someone else says '5' if we are talking about the same (it would lose its objective status) . And also if the mathematical entities had been mental entities, then how do we explain the relationship that exists, for example, between numbers and scientific theories, or that if there are infinitesimal numbers such as π, then if it is mental, there is no guarantee that it is So. If knowledge is born by abstraction, then it happens that mathematical knowledge is not a necessary knowledge, since it is contingent that my abstraction of objects from a series of numbers is correct. Frege puts his idea like this:
If the thought he expressed in the Pythagorean theorem can be recognized as true by others as well as by me, then it does not belong to the content of my consciousness; I am not, therefore, its bearer; however, I can recognize it as true. But if it is not the same thought that I or that other man consider to be the content of the Pythagorean theorem, then, strictly speaking, we should not say "the Pythagorean theorem", but "my Pythagorean theorem" or "his Pythagoras' theorem ”, and these would be different, since the meaning necessarily belongs to the sentence (Frege, 1996: 36).
In The Foundations of Arithmetic, Frege concludes that numbers are neither concrete 'external' entities nor mental entities of any kind. In "The Thought" he will give you the status of thoughts (the meaning of declarative sentences) and also, by implication, for their constituents, the meaning of the underlying expressions. Frege does not say that the senses are 'abstract', but that they belong to a third realm distinct from the external sensible world and the internal world of consciousness.
3. Why should I accept the existence of numbers?
W.V.O. Quine formulated in the 1960s a methodological critique of traditional philosophy, which he called naturalism, which defends that our best theories are scientific theories, so if we want to obtain the best available answer to the philosophical questions "how we know?" or "what entities exist?", we should appeal to scientific theories and not to traditional metaphysical theories, such as Berkeleylian empiricism or Hegelian idealism. For Quine, we should consult and analyze the best scientific theories to see what they tell us about the world, since they contain, implicitly, our best asset to determine what there is, what we know or how we know it.
However, approximately since Galileo, scientific theories put aside their expression in natural language, as happens in Aristotle's Physics, for a richer and more precise language, which is that of mathematics. Newton's theory of gravitation, quantum mechanics, chemical theories of reduction and oxidation reactions, ..., based on quantification on real numbers. Consequently, an ontological commitment to mathematical entities seems inherent in our best scientific theories.
Let's see the version of Quine's Argument from Mathematical Indispensability (AIMQ) as presented by Russell Marcus (n.d.) in the Internet Encyclopedia of Philosophy:
We should believe the theory which best accounts for our sense experience.
If we believe a theory, we must believe in its ontic commitments.
The ontic commitments of any theory are the objects over which that theory first-order quantifies.
The theory which best accounts for our sense experience first-order quantifies over mathematical objects.
We should believe that mathematical objects exist.
In AIMQ, premise (1) is the criterion of choice for our ontological commitment, based on the success of the theories, as well as on the two fundamental pillars of the Quinean criterion: naturalism and confirmatory holism. The thesis of confirmatory holism states that "individual sentences are only confirmed in the context of a broader theory" (Marcus, nd), so that theories can only be confirmed or refuted as one whole, since the meaning of certain scientific concepts only have felt within this theory, as is the notion of 'atom'.
Premise (2) is the Quinean criterion presented in a simplified way. It can be easily criticized from a deflationary position such as fictionalism, understanding that this commitment is pragmatically justified, it is not relevant in terms of assuming an ontology. Fictionalism understands theories as languages in which mathematical entities would be true-in-L, in the manner of a novel, but this position presupposes intensional notions such as synonymy or analyticity that a naturalist cannot afford given the denial of the discontinuity between philosophy and science, since it would be to reject confirmatory holism.
The premise (3) comes from the need for a regimented language that avoids obscure notions and treats predicates as if they were proper names.
The premise (4) is a premise common to the entire scientific community, because except epistemoxic anarchist conceptions such as those of Paul Feyerabend or pseudoscientific, they would not consider that mathematics is necessary for science as an essential language of their theories.
To reach the conclusion by (1) and (4) is due to the fact that Quine rejects the reductivism characteristic of neopositivism, since the naturalistic way of doing philosophy requires treating the data of the senses and scientific theories as one whole, and since the Mathematical entities, although they are not part of our sense data, they do constitute a fundamental part of scientific theories, fulfilling the principle of parsimony.
4. Benacerraf's dilemma
Following Liggis (2010) and Balaguer (2001), we can comment on a famous objection to mathematical Platonism: the Benacerraf Challenge. This challenge is about an epistemological problem proposed by Paul Benacerraf (1973) who maintains that mathematical entities do not exist since it is not possible to have knowledge of them. This problem is based on the fact that our best epistemological theory is the Causal Theory of Knowledge (CTK), which states that a subject s has knowledge of p if and only if s comes into contact with p. Let's see the argument (Balaguer, 2001: 22).
Human beings exist entirely in space-time.
If any abstract mathematical entity exists, then they exist outside of space-time.
(by CTK) If some abstract mathematical object exists, then human beings cannot gain knowledge of it
If mathematical Platonism is correct, then human beings cannot obtain mathematical knowledge
Humans have mathematical knowledge
Mathematical Platonism is not correct.
However, since the publication of "Mathematical truth" (1973), the causal theory of knowledge has been relegated as the best epistemological theory. Likewise, Liggins (2017: 68-69) recalls how Lewis (1986) inverted the argument in such a way that in reality, what Benacerraf's dilemma demonstrates is that mathematics refuted the causal theory of knowledge given a key evidence: that our knowledge mathematical is much more reliable than our knowledge of epistemology.
Here you can check some extended bibliography if you want:
Balaguer, M. (2001). Platonism and anti-platonism in mathematics. New York: OUP.
Benacerraf, P. (1973). Mathematical Truth. The Journal of Philosophy, 70(19), 661-679.
Bricker, P. (2016). Ontological Commitment. (E. N. Zalta, Ed.) Obtido de The Stanford Encyclopedia of Philosophy: <https://plato.stanford.edu/archives/win2016/entries/ontological-commitment/>.
Chalmers, D. J., Manley, D., & Wasserman, R. (Edits.). (2013). Metametaphysics. New essays on foundations of ontology. Oxford: OUP.
Cole, J. C. (s.d.). Mathematical Platonism. (J. Fieser, & B. Dowden, Edits.) Obtido de Internet Encyclopedia of Philosophy: <https://www.iep.utm.edu/mathplat/>.
Colyvan, M. (1998). In Defence of Indispensability. Philosophia Mathematica 6(1), 39-62.
Colyvan, M. (2019). Indispensability Arguments in the Philosophy of Mathematics. (E. N. Zalta, Ed.) Obtido de The Stanford Encyclopedia of Philosophy: <https://plato.stanford.edu/archives/spr2019/entries/mathphil-indis/>.
Eklund, M. (2006). Metaontology. Philosophy Compass 1/3, 317-334.
Frege, G. (1996). El pensamiento. En Pensamiento y lenguaje. Problemas en la atribución de actitudes proposicionales (págs. 23-48). México D.F.: UNAM Instituto de Investigaciones Filosóficas.
Horsten, L. (2019). Philosophy of Mathematics. (E. N. Zalta, Ed.) Obtenido de The Stanford Encyclopedia of Philosophy: <https://plato.stanford.edu/archives/spr2019/entries/philosophy-mathematics/>.
Lewis, D. (1986). On the Plurality of the Worlds. Oxford: Blackwell.
Liggins, D. (2010). Epistemological Objections to Platonism. Philosophy Compass 5/1, 67-77.
Lowe, E.J. (2002). A survey of metaphysics. Oxford: OUP.
Marcus, R. (s.f.). The Indispensability Argument in the Philosophy of Mathematics. (J. Fieser, & B. Dowden, Edits.) Obtenido de Internet Encyclopedia of Philosophy: <https://www.iep.utm.edu/indimath/>.
Quine, W.O.V. (2001). Palabra y objeto. Barcelona: Herder.
Quine, W.O.V. (2002). Desde un punto de vista lógico. Barcelona: Paidós.
Rosen, G. (2018). Abstract Objects. (E. N. Zalta, Ed.) Obtido de The Stanford Encyclopedia of Philosophy: <https://plato.stanford.edu/archives/win2018/entries/abstract-objects/>.