Abstract
From the perspective of mathematical practice, I examine positions claiming that mathematical objects are introduced by human agents. I consider in particular mathematical fictionalism and a recent position on social ontology formulated by Cole (2013, 2015). These positions are able to solve some of the challenges that non-realist positions face. I argue, however, that mathematical entities have features other than fictional characters and social institutions. I emphasise that the way mathematical objects are introduced is different and point to the multifaceted role that relations and interconnections play in this context. Finally, I argue that mathematical entities can be considered to be pragmatically real.
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Notes
This fact has been pointed out by others and expressed in different ways. I refer, for example, to Ferreirós (2016) who uses the terminology of ‘strata’ and ‘interconnections’.
I refer readers to Peirce's Prolegomena to an Apology for Pragmaticism from 1906 which is reproduced in (Peirce 1967) 4.530 - 4.572 and (Carter 2020) for an explanation of what is involved in necessary reasoning.
Ferreirós has explained how ‘hypothesis’ was used in this sense also by Riemann and Hilbert around the 19th century. Ferreirós (2016) writes that Riemann took a ‘hypothesis’ to be a supposition stressing that it is neither certain nor evident.
The reference (CP 4.132) to Peirce’s writings is the standard abbreviation for the Collected Papers edited by Hartshorne and Weiss (Peirce 1967) volume 4 and paragraph 132.
The way I characterise the two types of mathematical activity resembles — and is to some extent inspired by — Hilbert’s axiomatic method (Hilbert 1918). The first, consisting of deriving consequences from hypotheses, corresponds to Hilbert’s progressive development of a field of knowledge. The second part, of determining the underlying assumptions of a theory is referred to as deepening the foundation.
Today we would say that the natural numbers are characterised by Peano’s axioms. But if we consider the history of the natural numbers, or numbers more generally, one can find many different ways that they have been conceived and treated. See Chemla et al. (forthcoming) for interesting perspectives on the history of numbers. Chemla et al. point out a number of implicit assumptions in previous works on the history of numbers. Among them are the tendency to tell linear stories and to see the development as an evolutionary process culminating in a given conception of numbers and corresponding system of signs. In the light of the many different treatments of numbers throughout time I find that it is apt to characterise as ‘hypothetical’ any particular (including the current) postulates on numbers.
NE IV refers to the New Elements volume IV edited by Carolyn Eisele (Peirce 1976).
I refrain from explaining what ‘truth’ means in this context. I assume that it is fairly uncontroversial to say that if I have a collection of five birds and a collection of three birds and the two collections are disjoint, then I have a total of eight birds. It is more difficult to give an account of what is meant by the truth of a general proposition of mathematics. Note that I give an account of what it means to be a real substance in mathematics. I do not claim that the truth value of a mathematical proposition depends on whether it is reducible to statements concerning physical objects or not.
There are various versions of the argument for the claim that truth implies existence. The claim states that we commit ourselves to the existence of entities referred to in a statement if we say that it is true. I refer to Leng (2010) for an elaboration.
Thomasson distinguishes between different types of dependency relations. The distinctions are between an historical and constant dependency and rigid versus generic. The latter distinction resembles the difference between a token and a type. A work of art, for example, depends rigidly on the particular author that created it, but it is only generically dependent on competent readers. In addition, according to Thomasson, a work must have a constant readership for it to continue existing and so it is constantly dependent on there being a competent readership. The author, on the other hand, only needs to create it once at a certain point in time. The latter is referred to as ‘historical’ dependency.
The quote is taken from slides presented at the second Interepisteme workshop in Nice October 6, 2021.
A similar view is expounded in K. Devlin’s book The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip (2000). According to Devlin mathematics depends on what he refers to as “off-line thinking” which is the ability to reason abstractly in a conjectural, ‘what if’ mode. He further claims that mathematicians create a discourse about mathematical objects and their connections very much like the discourse used when we gossip about other people.
The square (formed by points) can be cut up into a smaller square formed on a side of \(c-1\) points and a gnomon that has \(2(c-1) +1\) points. When the last number is a square, one obtains a triple of squares that fulfil Pythagoras’ theorem.
Note that mathematical concepts might not have a very precise definition when first introduced and that precise definitions arise gradually. Thomas (2002) uses the term proto-mathematics (referring to Kitcher’s book The Nature of Mathematical Knowledge) to characterise parts of mathematics where concepts are not yet clearly defined. This could refer to our ancestors’ calculations with numbers and the first dealings with geometry. See also Ferreirós (2016) who explains how what he refers to as ‘technical practices’ develop into mathematical theories.
The proof goes like this: Suppose \(p=a\cdot b\). We need to show that either a or b is a unit. If \(p=a\cdot b\) then p divides \(a\cdot b\). According to the definition of being prime, this implies that p divides a or b. Assuming the first, it is the case that \(p\cdot k=a\) for some integer, k. Substituting \(a\cdot b\) for p, we see that \((ab)k=a\) and therefore that \(bk=1\). The latter implies that b is \(\pm 1\).
The example is taken from Peirce’s “Prolegomena for an Apology to Pragmatism” from 1906, published in (CP 4.530 – 572), see (Peirce 1967). In Peirce’s terminology, the map (and the pins in it) so used is a diagram, that is, an iconic representation of the battlefield and possible different configurations of soldiers.
I will not discuss these claims here. But note that Rytilä (2021) has recently argued that Cole’s position fails to account for the objectivity of advanced mathematics. The problem is that Cole introduces a layer in between the physical reality, the layer of “brute facts” and the layer(s) consisting of mathematical representations. The layer in the middle consists of ‘logical possible systems of objects’. Rytilä argues that Cole needs to explain what this intermediate level is and what the relation between mathematical entities and the brute facts of reality is before one can say anything about the objectivity and applicability of mathematics.
V denotes an algebraic manifold and W a vector bundle defined over it.
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Carter, J. Mathematical Practice, Fictionalism and Social Ontology. Topoi 42, 211–220 (2023). https://doi.org/10.1007/s11245-022-09856-4
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DOI: https://doi.org/10.1007/s11245-022-09856-4