## Abstract

Are there entities which are just distinct, with no discerning property or relation? Although the existence of such utterly indiscernible entities is ensured by mathematical and scientific practice, their legitimacy faces important philosophical challenges. I will discuss the most fundamental objections that have been levelled against utter indiscernibles, argue for the inadequacy of the extant arguments to allay perplexity about them, and put forward a novel defence of these entities against those objections.

This is a preview of subscription content, access via your institution.

## Notes

Since non-identity weakly discerns between the nodes in an edgeless graph, discerning relations must not involve identity. For more discussion on various modes of discernibility, see, for instance, Caulton and Butterfield (2012, §3). Our terminology for different modes of discernibility corresponds to Caulton and Butterfield’s paper.

The Principle of the Identity of Indiscernibles can be formulated using a second-order quantifier ranging over properties of the relevant philosophical significance: \(\forall P(Px \leftrightarrow Py)\rightarrow x=y.\)

Haecceitism is also a non-reductive account of identity, for according to haecceitism, indiscernibility is not sufficient for identity: two entities can be (qualitatively) indiscernible and yet, in terms of their haecceities, numerically distinct. But this does not mean that the proponent of utter indiscernibles is committed to haecceitism. She does not allow haecceities to discern between otherwise indiscernible entities.

Muller and Seevinck (2009) have argued that bosons in entangled states are weakly discernible.

I owe this point to Tim Button. Thanks to him here.

Shapiro formulates his minimalism constraint by restricting it to

*mathematical*properties. But he also seems to suggest that the constraint concerns the attribution of some non-mathematical properties, too. He writes:Speaking strictly, the thesis that mathematical structures are abstract, acausal, etc. is not part of mathematics. In a sense, mathematics leaves this bit of metaphysics open. To be sure, mathematicians do not attribute causal or other physical properties to mathematical objects – to places in structures – but on the view in question, one would not expect them to. A given mathematical structure might not have any physical properties, and even if it does have physical properties, a mathematician as such does not care about them. (Shapiro 2006, p. 119)

Since Shapiro finds a “‘quieter’ philosophy more pleasing” (2006, p. 112) so that he advices us not to attribute

*mathematical*properties unless those attributions are explicit or implicit in mathematics, he will have every motivation to give a more generous formulation of his minimalism constraint: do not attribute*any*properties to mathematical objects unless those attributions are explicit or at least implicit in mathematics itself.The point concerning two different conceptions of unlabelled graphs has been discussed by De Clercq (2012, p. 666). For references to graph theory textbooks, see also De Clercq (2012, p. 668). Button and Walsh (2018, pp. 372–373) have also criticized Leitgeb and Ladyman’s appeal to graph-theoretic practice on the basis of the double life of unlabelled graphs.

This conception of an object corresponds to what some writers call ‘individual’. An individual, thus construed, is an entity which is discernible, either intrinsically or extrinsically, from other elements. For further discussion and references, see Caulton and Butterfield (2012, §3.2).

For a defence of this objection from circularity, see, for example, MacBride (2006).

For comments and discussion, I am very grateful to Tim Button, Simon Hewitt, Øystein Linnebo, Jonathan Nassim, Jack Woods, and anonymous referees.

## References

Benacerraf, P. (1965). What numbers could not be.

*Philosophical Review*,*74*(1), 47–73.Black, M. (1952). The identity of indiscernibles.

*Mind*,*61*, 153–164.Button, T. (2006). Realistic structuralism’s identity crisis: A hybrid solution.

*Analysis*,*66*(3), 216–222.Button, T. (2017). Grades of discrimination: Indiscernibility, symmetry, and relativity.

*Notre Dame Journal of Formal Logic*,*58*(4), 527–553.Button, T., & Walsh, S. (2018).

*Philosophy and model theory*. Oxford: Oxford University Press.Cantor, G. (1895).

*Contributions to the founding of the theory of transfinite numbers*(P. E. B. Jourdain, Trans.). New York: Dover.Caulton, A., & Butterfield, J. (2012). On kinds of indiscernibility in logic and metaphysics.

*British Journal for the Philosophy of Science*,*63*(1), 27–84.De Clercq, R. (2012). On some putative graph-theoretic counterexamples to the principle of the identity of indiscernibles.

*Synthese*,*187*(2), 661–672.Frege, G. (1892). Rezension von: Georg Cantor, Zur Lehre von Transfiniten. Gesammelte Abhandlungen aus der Zietschrift für Philosophie und Philosophische Kritik.

*Zietschrift für Philosophie und Philosophische Kritik*,*100*, 269–72. Page reference is to Frege (1967).*Kleine Schriften*. Hildenscheim: Olms.Frege, G. (1899). On Mr. H. Schubert’s numbers. In B. McGuinness (Ed.),

*Collected papers on mathematics, logic and philosophy*(pp. 249–272). New York: Basil Blackwell.Hallett, M. (1984).

*Cantorian set theory and limitation of size*. Oxford: Clarendon Press.Hellman, G. (2004). Russell’s absolutism vs. (?) structuralism. In G. Link (Ed.),

*One hundred years of Russell’s Paradox: Mathematics, logic, philosophy*(pp. 249–272). Berlin: Walter de Gruyter & Co.Hellman, G. (2005). Structuralism. In S. Shapiro (Ed.),

*The Oxford Handbook of philosophy of mathematics and logic*(pp. 536–562). Oxford: Oxford University Press.Keränen, J. (2001). The identity problem for realist structuralism.

*Philosophia Mathematica*,*9*(3), 308–330.Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology.

*Philosophia Mathematica*,*16*(3), 388–396.MacBride, F. (2006). What constitutes the numerical diversity of mathematical objects?

*Analysis*,*66*(289), 63–69.Muller, F. A., & Seevinck, M. (2009). Discerning elementary particles.

*Philosophy of Science*,*76*(2), 179–200.Quine, V. V. O. (1970).

*Philosophy of logic*. Cambridge, MA: Harvard University Press.Saunders, S. (2003). Physics and Leibniz’s principles. In K. Brading & E. Castellani (Eds.),

*Symmetries in physics: Philosophical reflections*. Cambridge: Cambridge University Press.Saunders, S. (2006). Are quantum particles objects?

*Analysis*,*66*(289), 52–63.Shapiro, S. (1997).

*Philosophy of mathematics: Structure and ontology*. New York: Oxford University Press.Shapiro, S. (2006). Structure and identity. In F. MacBride (Ed.),

*Identity and modality*(pp. 34–69). Oxford: Oxford University Press.Shapiro, S. (2008). Identity, indiscernibility, and

*ante rem*structuralism: The tale of $i$ and $-i$.*Philosophia Mathematica*,*16*(3), 285–309.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Assadian, B. In defence of utterly indiscernible entities.
*Philos Stud* **176**, 2551–2561 (2019). https://doi.org/10.1007/s11098-018-1140-5

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11098-018-1140-5

### Keywords

- Identity
- Indiscernibility
- Utter indiscernibility
- Naturalism
- Objecthood