What Cannot Be the Rationals, the Irrationals and Other Riddles


This article aims to show that unless we consider Zeno’s paradoxes in the original metaphysical perspective in which they were generated, any attempt at understanding, let alone solving them, is destined to fail. This perspective, I argue, is the dichotomy of One and change. These latter were defined at the outset of Western philosophical thought by Parmenides as the two paths of the rational, i.e. accountable by a self-identical thought and thus real (One), and the non-identical change, irrational and unreal. In this perspective, the irrational, is by definition unnameable (alogos) and thus uncountable. I claim that we have inherited this dichotomic thought and if we become aware of this legacy, many deadlocked paradoxes or logical aporias in Western epistemology will acquire the status of logical necessities that follow directly from this dichotomy.

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  1. 1.

    Papa-Grimaldi, A., (1996) “Why Mathematical Solutions to Zeno’s Paradoxes Miss the Point” Review of Metaphysics, Vol. 50. pp.299-314

  2. 2.

    See for example M. Zangari (1994), ‘Zeno, Zero and Indeterminate Forms: Instants in the Logic of Motion’, Australasian Journal of Philosophy, Vol. 72., pp.187-204;

    W. I. McLaughlin, (1994) “Resolving Zeno’s Paradoxes”, Scientific American, November, pp.66-71.

  3. 3.

    See Carl Boyer The History of Calculus and its Conceptual Development Dover Publications p. 295. The author is one of a long list of historians and mathematicians who hold that Zeno’s paradoxes are simply mathematical problems for which modern calculus provides a mathematical solution. The champion of this trend was of course B. Russell who in his The Problem of Infinity Considered Historically smugly dismisses Zeno’s paradoxes as reliant on ‘prejudices instilled by the arithmetic learnt in childhood’

  4. 4.

    I do not wish to address directly these interpretations, such as the one provided by P. Lynds*, though the arguments developed in the paper are designed to answer the most frequent misunderstandings and blunders related to Zeno’s paradoxes.

    *Lynds, Peter (2003) “Zeno's Paradoxes: A Timely Solution”. PhilSci Archive, URL: http://philsci-archive.pitt.edu/id/eprint/1197 (accessed 2012-02-15).

    Lynds, Peter (2003) “Time and Classical and Quantum Mechanics: Indeterminacy Versus Discontinuity”, Foundations of Physics Letters 16, no. 4: 343-355

  5. 5.

    For a detailed exposition and a systematic treatment of the paradoxes, see Wesley C. Salmon ed., Zeno's

    Paradoxes (Indianapolis: Bobbs-Merrill, 1970).

  6. 6.

    ‘…but also from this, on which mortals who know nothing wander, double-headed; for helplessness guides the wandering thought in their hearts. They are carried deaf and blind at the same time, amazed, a horde incapable of judgment, by whom to be and not to be are considered the same and yet not the same, for whom the path of all things is backward turning’ p.54 Taran, L. (1965) Parmenides, Princeton, New Jersey, Princeton University Press.

  7. 7.

    It will be useful to recall in this connection that many centuries afterwards, but following a similar reasoning, Augustine talking about the essence of time will say that it “tends towards non-being” (Augustine, Confessions ch. XIV)

  8. 8.

    In fact, the consequences of the dichotomy of one and many are to be seen also in the ethical dimension as argued in the book Orphans of the One. But here we will limit our enquiry to the epistemological level. Papa-Grimaldi, A., (2010) Orphans of the One or the Deception of the Immanence: Essays on the Roots of Secularization Peter Lang, Berne).

  9. 9.

    See Papa-Grimaldi, A., (2010) Orphans of the One, p.39

    To be more precise we should say that this “beginning” is often conceived as anterior to the One itself, from which the many are generated. Thus we read in the first verses of Daodejing: “The Tao gave birth to the One, the One gave birth to Two, the Two gave birth to Three. The three gave birth to all creation.” Similarly in Speusippus, a Pythagorean and member of the platonic academy, we read that the One is a perfectly simple and transcendental Being. This supreme One, principle and unity of all things, combined with multiplicity, produces a secondary One or Unit (see fn 15 in Essay I Orphans of the One). About the distinction between the One as absolute principle and Unity as principle of multiplicity, also Plotinus writes extensively in The Enneades. (VI, 2,9.) See of course also Plato’s Parmenides (142 c-d).

  10. 10.

    This beginning or principle of everything which metaphysics and religion have always treated with the utmost awe, has become in our times the controversial object of physical knowledge in the theory of the physical forces at work in our universe, and at the beginning of time and space (see for example A Universe from Nothing. Lawrence M. Krauss and Origins: Fourteen Billion Years of Cosmic Evolution. Neil deGrasse Tyson).

    Thus, the passage formalized and forbidden by Parmenides, from identity to movement, from a self-identical being, to temporal becoming should lose its ineffability and become itself object of enquiry. Whether physics from within the western epistemic tradition shaped by the Parmenidean dichotomy, will be able to avoid the inner logical contradiction of a passage from pure being or pure nothing to self-differentiation which has been the Achilles’ heel of every metaphysics, is a matter that remains, by definition, entirely of a philosophical nature.

  11. 11.

    A similar argument can be found for example in “The Bhagavad Gita”: “That which is non-existent never comes into being, that which is existent never goes to non-being” (ch 2 verse 16) Could it be possible to affirm, following Ferenc Ruzsa’s suggestion, that Parmenides borrowed his conception of Being from the Upanisads, during a journey to India where he became acquainted with the language and the philosophy of that country?

  12. 12.

    258c-d in The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns (Princeton University Press, 1961

  13. 13.

    pan d’empleon estin eontos (PARMENIDES' FRAGMENT VIII Diels-Kranz verse 24

  14. 14.

    This Being will be the prototype of the eternal now of God in successive speculation, it will become the “interminabilis vitae tota simul atque perfecta possessio” of Boethius (De Consolatione Philosophiae v69-11.) Aquinas who will follow Boethius and Aristotle will also conceive eternity as one and simultaneous: whilst time contains a before and after, “aeternitas est tota simul”. Also Augustine will oppose the unchanging, aeternal present of God to the constant movement of Time “non autem praeterire quicquam in aeterno, sed totum esse praesens; nullum vero tempus totum esse praesens” Confessions XI 11). This is the necessary tautology that will never be forgotten by the West. It is tautologically true for a thought shaped by the dichotomy of identity and change, to deny being and presence to the non-identical change of time ‘which is-not’, by definition, for it is changing all the time and it is thus unthinkable, un-nameable, a-logos.

  15. 15.

    Fragment 88, Heraclitus, Fragments (Pre-Socrates, Vol. 2) T.M. Robinson (Editor) University of Toronto Press, Scholarly Publishing Division (July 11, 1987)

  16. 16.

    See for more details on the subject: History of Mathematics, Volumes I and II, D.E. Smith, 1953 and The Irrationals: A Story of the Numbers You Can’t Count on, Julian Havil, 2012.)

  17. 17.

    See also Russell, B. (1961) History of Western Philosophy, London, George Allen & Unwin., p.63

  18. 18.

    See also Essays II and III in Papa-Grimaldi, A., (2010) Orphans of the One or the Deception of the Immanence, Peter Lang, Bern

  19. 19.

    Aristotle Physics VI:9, 239b5

  20. 20.

    It is only with the first metaphysical attempt to mediate the One and the many, the “tormented” Platonic theory of participation, that concept appears of an ideal or logical arrow which can be thought independently of the constant physical change and should thus allow the knowledge of the “unstable” physical arrow, which Parmenides had declared unknowable. Plato is still aware of the insuperable dichotomy, for his fundamental belief is, as we read in Aristotle, that “all sensible things are ever in a state of flux and there is no knowledge about them”. (Metaphysics, book I). However, our reflections, here, also intend to retrospectively illustrate why such attempts at reconciliation were all destined to fail, in the face of the tautological force of the original dichotomy.

  21. 21.

    At this point in western thought the one and the many, identity and change still possess an ontological value and thus appear incommensurable. It will be the work of metaphysical thought to overcome the dichotomy of identity or the real vs many or the unreal, by reducing the identity from an ontological to a logical feature of our thought, first with Plato’s Sophist and then in a way that will consign the aporia to the oblivion of an ancient conundrum, with Aristotle’s logic. In Aristotle we find the Parmenidean ontological identity formalized in the first law of reason: the law of identity. Thanks to the identity as a formal law, thought will be able to accept change in a logical discourse, though it will never overcome the chasm between the identical and the ever changing, once they have been separated by Parmenides. However, with Aristotle’s logical and metaphysical work, the dichotomy is overcome and “forgotten”. For a deeper understanding of this process and its undying consequences in Western Thought, see Alba Papa-Grimaldi, (2010) Orphans of the One or the Deception of the Immanence Peter Lang, Bern.

  22. 22.

    See also “The ontologization of movement in Aristotle” in Papa-Grimaldi (2010) Orphans of the One, pp.109-112

  23. 23.

    Thus we can see that, despite the fact that the seventeenth century hails the project of a mathesis universalis, it is precisely in one of the first champions of the mathematization of the Universe that such difficulty is clearly recorded. In a letter to Mersenne(15 April 1630) Descartes wrote:

    As for Problems, I would send you a million to propose to others if you desire it; but I am so tired of mathematics and now hold it in so low esteem that I could no longer bother to solve them myself (See Andrews p.55)

    Andrews, Floy E. (1998) “God, the Evil Genius and Eternal Truths: The Structure of the Understanding in the Cartesian Philosophy.” Animus, 3 http://www2.swgc.mun.ca/animus/vol3.html (Accessed 15 Feb, 2012).

    As he became increasingly aware of the fact that mathematical certainty could not ground the empirical element and, thus, grant certainty to our knowledge of the changing world. It is furthermore symptomatic that such awareness of the incommensurability of the One of thought and the many of change, resurfaces in the very thinker who after many centuries of constructive metaphysics, has unearthed with the methodological doubt the “murderous” roots of a metaphysics of reconciliation. Foundational thought, which in Plato is already described as a “parricide” of the father Parmenides and of his prohibition to bridge the One and the many, is put under scrutiny by Descartes whose “doubt”, once again, will stop only at the One of thought, in the form of the Cogito: the identity of thought with itself, devoid of any other content.

  24. 24.

    Bishop Berkeley, as it is well known, was the first to oppose a metaphysical critique to these concepts. Berkeley, G. (1734), The Analyst, ed. D.R. Wilkins, online e-text at < http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst > Accessed 12 August 2013.

  25. 25.

    see also Papa-Grimaldi, A. (2007), The Presumption of Movement Axiomathes Vol 17, pp137-154.

  26. 26.

    Plato, Parmenides, 139c3-dl)

  27. 27.

    These actual words, to be precise, are not to be found in any of Heraclitus fragments, they were rather the slogan of his commentators to give a synthetic expression to his belief in reality as constant becoming and as opposed to the Eleatic immobile being. See fragment 91 Diels-Kranz)

  28. 28.

    The Critique of Pure Reason. Second Analogy A209

  29. 29.

    Augustine, Confessions ch. XIV.

  30. 30.

    I will not, here, concern myself with the exegetical debate on the sense of the second part, the cosmological part, of the Parmenidean poem; whether the cosmological theories that he exposes there amounted for Parmenides to pure folly, or to a knowledge which is not as real as that of being, but mere “opinion of the mortals”, is a question that remains open amongst scholars. However, the outcome of this debate has no bearing on the thesis of this article, for it remains certain that the Parmenidean ontology is founded on the identity of Being.

  31. 31.

    See for further insights on this issue, A. Papa-Grimaldi, Orphans of the One, Essay I

  32. 32.

    See also for an interesting insight in this problems M. Eldred The Digital Cast of Being, http://www.arte-fact.org/dgtlon_e.html, Accessed 12 August 2013

  33. 33.

    See also in this respect the interesting remarks by U. Mohrhoff, “The one, the many, and the quantum” arXiv:quant-ph/0005110 (2000).

  34. 34.

    For further insight in this argument, see Papa-Grimaldi, A. (1998), Time and Reality, Aldershot, Ashgate Publishing. pp74-79

  35. 35.

    In this light, how could the suggestion of P. Lynds* be considered a solution of Zeno’s paradoxes since it ignores precisely this pre-requisite of them? To say ‘that physical entities cannot be considered at an instant because they never stand still’, is in this light the discovery of hot water.

    *Lynds, Peter (2003) “Zeno's Paradoxes: A Timely Solution”. PhilSci Archive, URL: http://philsci-archive.pitt.edu/id/eprint/1197 (accessed 2012-02-15).

    Lynds, Peter (2003) “Time and Classical and Quantum Mechanics: Indeterminacy Versus Discontinuity”, Foundations of Physics Letters 16, no. 4: 343-355

  36. 36.

    I will just quote here the Aristotelian rendering of Zeno’s paradoxes that challenge motion if we start from the many. The Achilles states:

    In a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead - Aristotle, Physics VI:9, 239b15

    Similarly the dichotomy states:

    That which is in locomotion must arrive at the half-way stage before it arrives at the goal - Aristotle, Physics VI:9, 239b10

    I agree with Aristotle that they are substantially the same paradox(See section d of this paper) and I refer the reader for a detailed discussion of them to a classic of the literature on Zeno’s paradoxes: Wesley C. Salmon ‘Zeno’s Paradoxes’ (Indianapolis: Bobbs-Merril, 1970)

  37. 37.

    For an extended discussion of the so called “Standard Solution” see also The Standard Solution to the Paradoxes, The Internet Encyclopedia of Philosophy, http://www.iep.utm.edu/zeno-par/#H2 Accessed 20 April 2014

  38. 38.

    For more details on the discovery of the irrationals, see also: The Irrationals: a Story of the Numbers You Can’t Count On, Julian Havil, Princeton University Press (July 22, 2012)

  39. 39.

    see also Papa-Grimaldi, A., (2010) Orphans of the One or the Deception of the Immanence Peter Lang, Bern, p.63

  40. 40.

    Plato, 1961, 128a-c, p.923 in The Collected Dialogues of Plato, tr. E. Hamilton & H. Cairns, (New Jersey:Princeton University Press, 1961), p.923.

  41. 41.

    Plato, 1961,, 128a-c, p.923

  42. 42.

    Plato, 1961, 128 a-c, p.923

  43. 43.

    See Papa-Grimaldi, A. (1996b) “The Paradox of Phenomenal Observation”, Journal of the British Society for Phenomenology, 27, no. 3: 294-312.

  44. 44.

    ‘In the [field of] appearance there is no difference of the real that is the smallest, just as in the magnitude of times there is no time that is the smallest.’ Kant, I. (1956) p. 231. [Kant, I. (1956), Critique of Pure Reason tr. N. K. Smith, London, Macmillan & Co. Ltd.]

  45. 45.

    See for more details Papa-Grimaldi, A. (2007), The Presumption of Movement Axiomathes Vol 17, pp137-154.

  46. 46.


  47. 47.

    In this connection see also Papa-Grimaldi, A. (1996b) “The Paradox of Phenomenal Observation”, Journal of the British Society for Phenomenology, 27, no. 3: 294-312.

  48. 48.

    Papa-Grimaldi, A., (2010) Orphans of the One or the Deception of the Immanence, Peter Lang, Bern

  49. 49.

    …that queer thing the instant. The word instant appears to mean something such that from it a thing passes to one or other of the two conditions....there is no transition from a state of rest…nor from motion so long as it is still in motion, but this queer thing, the instant is situated between the motion and the rest; it occupies no time at all…

    Plato 156d-e, in Plato, (1999), The Collected Dialogues of Plato, ed. Edith Hamilton and Huntington Cairns, Princeton, Princeton University Press

  50. 50.

    See for further development of this idea Papa-Grimaldi, A. (1996b) “The Paradox of Phenomenal Observation”, Journal of the British Society for Phenomenology, 27, no. 3: 294-312.

  51. 51.

    Kant, I. (1956) p. 231.

  52. 52.

    The outcome of this evolution is not trivial, since the moral crisis of the West, which seems to this day to have no positive resolution, has its roots in our peculiar epistemic situation that, since Parmenides, has deprived the world of change of its foundation and legitimacy. The subsequent evolution of our thought from foundational metaphysics into the mathematization of reality, has made us even more, “constitutionally” incapable of harmonizing the various forms of knowledge of the many. For more insights into this, I refer the reader to Orphans of the One

  53. 53.

    Hegel., The Science of Logic. pp. 94-5. Hegel, G. W. F. (1969) Science of Logic tr. A. V. Miller, (Atlantic Highlands, NJ:Humanities Press International, Inc.).

    Our constructive metaphysics and our scientific method that builds a bridge between the One and the many to overcome the illusory nature of the world of change, is one side of the coin; the other side is the philosophical and religious experience of Hinduism and Buddhism that have accepted the irreality of the impermanent and thus have not created a world, like the western one, found on the attempted and failed reconciliation of the One and many, an attempt which with all its tensions and paradoxes is nonetheless the essence of our culture and of all its achievements for better and for worst.

  54. 54.

    For a further understanding of the retroactive influence of the dichotomy on our philosophical and scientific thought, I refer the reader to A. Papa-Grimaldi, Orphans of the One

  55. 55.

    In the Cratylus Plato argues against Heraclitus as follows: “How can that be a real thing which is never in the same state?…for at the moment that the observer approaches, then they become other….so that you cannot get any further in knowing their nature or state…but if that which knows and that which is known exist ever…then I do not think it can resemble a process or flux….” Cratylus Paragraph 440 sections c-d)

  56. 56.

    Parmenides, On Nature, 4,5

  57. 57.

    Aristotle, Metaph. 1.10 (993a11-16)


I would like to thank Dr Michael Eldred for kindly exchanging views with me on an earlier version of this paper.

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Papa-Grimaldi, A. What Cannot Be the Rationals, the Irrationals and Other Riddles. Philosophia 43, 153–174 (2015). https://doi.org/10.1007/s11406-014-9562-6

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  • Paradoxes
  • Zeno
  • Parmenides
  • Reality
  • Quantum physics
  • Time and reality
  • Metaphysics