Abstract
In this paper, I explore Bunge’s fictionism in philosophy of mathematics. After an overview of Bunge’s views, in particular his mathematical structuralism, I argue that the comparison between mathematical objects and fictions ultimately fails. I then sketch a different ontology for mathematics, based on Thomasson’s metaphysical work. I conclude that mathematics deserves its own ontology, and that, in the end, much work remains to be done to clarify the various forms of dependence that are involved in mathematical knowledge, in particular its dependence on mental/brain states and material objects.
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Notes
- 1.
It seems that the first implicit expression of this claim came in the first volume of Bunge’s Treatise (See Bunge 1974). The detailed exposition appeared in the first part of the seventh volume of the Treatise (See Bunge 1985). The fundamentals of the position have not changed since then (See Bunge 1997, 2016).
- 2.
In the present paper, ‘mathematics’ always refer to contemporary mathematics, for this is what Bunge has in mind in his analysis.
- 3.
It could certainly be said that art should not try to be. An interesting question is where philosophy stands in this framework. Philosophy does not have the same conceptual autonomy as mathematics.
- 4.
This is not to say, of course, that mathematics and art have nothing in common. Historically, mathematics has been associated to a technè and I, for one, have argued that a large part of contemporary mathematics should be thought of as a systematic technology (Marquis 1997, 2006). I am here concentrating on the idea that mathematical objects and certain artistic objects, mostly literary ‘objects’, should be subsumed under the ontological category of fictions.
- 5.
Already in 1981, before the publication of Bunge’s volume 7 of the Treatise, Roberto Torretti had already identified three different kinds of mathematical fictionalism (Torretti 1981). I must confess that I do not understand his classification and will therefore refrain from using it. His claim that Bunge’s position might, in the end, be a form of idealism, is, however, not ridiculous. See also Robert Thomas’ excellent papers on fiction and mathematics (Thomas 2000, 2002).
- 6.
In this particular regard, Bunge’s position is not very different from what is now called ‘mathematical fictionalism’ in the literature. More about this link or, to be more exact, its absence, in the next section.
- 7.
The latter criterion is at the source of the vast literature on mathematical fictionalism. (See, for instance, Field (1980, 1989), Balaguer (1998), Yablo (2002), and Leng (2009)). Indeed, this criterion together with the so-called Quine-Putnam indispensability argument, seemed to provide good reasons for a certain form of Platonism with respect to mathematical objects. In this context, the claim that mathematics is a fiction is taken to follow from the claim that mathematics, like fiction, is not literally true, precisely because in both cases, these discourses literally fail to refer. Bunge has always resisted these Quinian arguments and he also very quickly pushed aside these fictionalist strategies, which he considers to be forms of nominalism and finds inadequate. Lately, Quine’s arguments have been criticized and therefore the motivation for this form of mathematical fictionalism has somewhat shifted. See, for instance, Thomasson (2014).
- 8.
Bunge is of course well aware that we cannot prove the consistency of most of our mathematical theories, in particular set theory and for a foundational categorical theory. Should we conclude that we simply cannot know that, in the end, our mathematical constructs exist?
- 9.
It is well known that Cantor and Russell resisted the introduction of infinitesimals for purely ideological reasons, even when they were perfectly acceptable objects in algebra at the time (See Ehrlich 2006).
- 10.
Of course, that is one of the reasons why art is so powerful: even though we know that it is all a pretense, we feel emotions just as strongly as when it is real. Some people simply cannot watch horror movies, although they know they are watching movies, i.e. fictions.
- 11.
That seems to be an easy exercise in Bunge’s framework. If we were to feign that mathematical objects really existed, then it means that mathematical objects could be in various states. What exactly these states would be, that would have to be determined. Would they be more like physical objects or living organisms? It is up to your imagination to decide. I suspect that in some cases, the ‘reality’ would be expressed more in terms of an independence from the mind, the will of the subject, in contrast with the objects that we create. But this shows, once again, that if we pretend that mathematical objects really exist, we do it in a very selective fashion without having learned anything about it.
- 12.
Thomasson seems to have move away from the specifics of her earlier theory. I stick to it simply because it provides an ontological analysis of fictions as dependent entities, thus an analysis that is close to Bunge’s claims. I am not claiming that it is the most adequate analysis. In fact, I would be inclined to address these issues more in the spirit of Thomasson’s recent work. That is another matter.
- 13.
Thomasson offers an interesting and rich classification of artefacts based on certain properties of the dependence relation. We refer the reader to her book for more.
- 14.
Otavio Bueno has sketched a form of mathematical fictionalism based on Thomasson’s views that is strikingly close to Bunge’s. Bueno defends the idea that mathematical entities are like fictional characters since, according to him, they are created in a particular context and in a particular time and their existence depends upon the existence of written papers and competent readers. He even adopts an existence predicate and distinguishes it from the existential quantifier. However, in the end, his position differs both from Bunge’s position and from Thomasson’s. We cannot do it justice in such a short paper. See Bueno 2009.
- 15.
Thomasson argues that the relations of dependence, constant dependence and historical dependence are all reflexive and transitive. This suggests that the resulting ontology could be formalized using the mathematical theory of categories, by representing the relation of dependence by a morphism between objects. The links between the kinds of dependence can be represented by functors. In fact, the distinction between rigid dependence and generic dependence can also be captured via a specific type of adjunction. This is not surprising given the fact that mathematical functions capture a form of dependence. We even talk about dependent and independent variables. We leave this project for another paper.
- 16.
We will stick to the terminology of mental states instead of brain states, despite the fact that we are in a materialist ontology. The reason for this choice is that the term ‘mental states’ already suggests a certain independence from particular brains but still indicates a clear and well-understood realm of discourse. Of course, the question as to how mental states depend on brain states is fundamental and, at some point, we might be ready to talk about brain states. See, for instance, Piazza and Izard 2009. By saying that constructs are equivalence classes of brain states, Bunge himself introduces a different identity criterion for mental states than for brain states, thus introducing the possibility of treating them as a genuine category.
- 17.
Thomasson herself distinguishes two opposites: the mental-material and the real-ideal. The first one would be reflected in the space of mental dependence, the material being independent of anything mental, and the second one would be placed in the space of spatiotemporal dependence, the ideal – which, from a Platonic point of view, would include numbers and similar entities – being independent of anything real. Notice that the material and the ideal are thus characterized purely negatively (Thomasson 1999, p. 125).
- 18.
The reader might want to include logic here. That is another issue. For a long time, this conviction was captured by the claim that mathematics is analytic.
- 19.
One interesting exception is Erhard Scheibe who, in his tribute to Bunge in 1981, presents an analysis of invariance and covariance of physical theories based on Bourbaki’s analysis. See Scheibe 1981.
- 20.
References
Balaguer, M. (1998). Platonism and antip-platonism in mathematics. New York: Oxford University Press.
Bueno, O. (2009). Mathematical fictionalism. O. Bueno, ∅. Linnebo, New waves in philosophy of mathematics (59–79). London: Palgrave Macmillan.
Bunge, M. (1974). Treatise on basic philosophy. Vol. 1. Semantics I: Sense and reference. Dordrecht: Reidel.
Bunge, M. (1977). Treatise on basic philosophy. Vol. 3. Ontology I: The furniture of the world. Dordrecht: Reidel.
Bunge, M. (1985). Treatise on basic philosophy. Vol. 7. Philosophy of science and technology part 1. Dordrecht: Reidel.
Bunge, M. (1997). Moderate mathematical fictionism. In E. Agazzi & G. Darwas (Eds.), Philosophy of mathematics today (pp. 51–71). Boston: Kluwer Academic.
Bunge, M. (2006). Chasing reality: Strife over realism. Toronto: University of Toronto Press.
Bunge, M. (2016). Modes of existence. Review of Metaphysics, 70, 225–234.
Dehaene, S. (2011). The number sense (2nd ed.). New York: Oxford University Press.
Dehaene, S., & Brannon, E. M. (2011). Space, time and number in the brain: Searching for the foundations of mathematical thought (Attention and performance, Vol. 24). London: Academic.
Dillon, M. R., Hu ang, Y., & Spelke, E. S. (2013). Core foundations of abstract geometry. Proceedings of the National Academy of Sciences, 110, 14191–14195.
Ehrlich, P. (2006). The rise of non-Archimedean mathematics and the roots of a misconception I: The emergence of non-Archimedean systems of magnitudes. Archives for the History of Exact Sciences, 60, 1–121.
Field, H. (1980). Science without numbers. Princeton: Princeton University Press.
Field, H. (1989). Realism, mathematics and modality. Oxford: Blackwell.
Leng, M. (2009). Mathematics and reality. Oxford: Oxford University Press.
Makkai, M. (1998). Towards a categorical foundation of mathematics. In J. A. Makowsky & E. Ravve (Eds.), Logic colloquium ‘95: Proceedings of the annual European summer meeting of the ASL, Haifa, Israel (pp. 153–190). Berlin: Springer.
Makkai, M. (2013). The theory of abstract sets based on first-order logic with dependent types. http://www.math.mcgill.ca/makkai/Various/MateFest2013.pdf
Marquis, J.-P. (1997). Abstract mathematical tools and machines for mathematics. Philosophia Mathematica, 5(3), 250–272.
Marquis, J.-P. (2006). A path to the epistemology of mathematics: Homotopy theory. In J. Gray & J. Ferreiros (Eds.), The architecture of modern mathematics (pp. 239–260). Oxford: Oxford University Press.
Marquis, J.-P. (2011). Mario Bunge’s philosophy of mathematics: An appraisal. Science & Education, 21(10), 1567–1594.
Marquis, J.-P. (2012). Categorical foundations of mathematics: Or how to provide foundations for abstract mathematics. The Review of Symbolic Logic, 6(1), 51–75.
Marquis, J.-P. (2018). Unfolding FOLDS: A foundational framework for abstract mathematical concepts. In E. Landry (Ed.), Categories for the working philosopher (pp. 136–162). New York: Oxford University Press.
Piazza, M., & Izard, V. (2009). How Humans Count: Numerosity and the Parietal Cortex, in Neuroscientist, 15(3), 261–273.
Scheibe, E. (1981). Invariance and covariance. In J. Agassi & R. S. Cohen (Eds.), Scientific philosophy today (pp. 311–332). Dordrecht: Reidel.
Thomas, R. (2000). Mathematics and fictions I: Identification. Logique et Analyse, 43(171–172), 301–340.
Thomas, R. (2002). Mathematics and fictions II: Analogy. Logique et Analyse, 45(177–178), 185–228.
Thomasson, A. (1999). Fiction and metaphysics. Cambridge: Cambridge University Press.
Thomasson, A. (2014). Ontology made easy. New York: Oxford University Press.
Torretti, R. (1981). Three kinds of mathematical fictionalism. In J. Agassi & R. Cohen (Eds.), Scientific philosophy today: Essays in honor of Mario Bunge (pp. 399–414). Boston: Reidel.
Yablo, S. (2002). Go figure: A path through Fictionalism. Midwest Studies in Philosophy, 25, 72–102.
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The author acknowledges the support of the SSHRC of Canada given for completion of this work.
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Marquis, JP. (2019). Bunge’s Mathematical Structuralism Is Not a Fiction. In: Matthews, M.R. (eds) Mario Bunge: A Centenary Festschrift. Springer, Cham. https://doi.org/10.1007/978-3-030-16673-1_33
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