Written By Thomas Perez. August 10, 2010 at 6:37AM. Copyright 2010.
Introduction – A Continuation From Part 1
The great German mathematician Abraham Robinson reasoned similarly, confessing “I cannot imagine that I shall ever return to the creed of the true Platonist, who sees the world of the actual infinite spread out before him and believes that he can comprehend the incomprehensible.”24
What else might be wrong with (1b)? No doubt the most frequent argument lodged against (1b) is known as the epistemological objection.25 If abstracta are independently existing realities as (1b) holds and are also causally effete (as no philosopher to my knowledge denies), then it would be impossible for us to have knowledge of them. To have knowledge of something independent of us entails a relation of sorts; one where information about some object of knowledge can pass from it to us or vice versa. But how can this be if abstracta exist a se and extended? (1b) therefore renders abstracta epistemologically inaccessible. Where O is some object of knowledge (say, an abstract object), the argument can be summarized:
(2b.1) If O is external to S, S can have knowledge of O only if there is some causal relation R between S and O
(2b.2) O is such that it cannot enter R
(2b.3) If (1b) is true, then O is external to S
(2b.4) Therefore if (1b) is true, then S cannot have knowledge of O
(2b.5) But S has knowledge of O
(2b.6) Therefore, (1b) is false
The best escape route for the platonist would be to refute (2b.1) by offering an account of how we can have knowledge of O that does not involve R. “After all,” the platonist might insist, “the very origin of the term ‘abstract object’ comes from their being ‘abstracted’ from our concepts.” Two replies can be made. First, so long as these “abstractions” remain derivatives of our concepts and do not somehow become independent, free-floating entities, the conceptualist will welcome this view as being similar to his own. Second, so reducing abstract objects to epistemic affairs only is a common nominalist strategy entirely inconsistent with plationism. So what could the platonist suggest as a non-causal relation between us and abstracta? Some form of intuition will be the most likely candidate here, serving to grant us access to abstracta in much the same way perception does concreta. Maybe our minds are such that we simply grasp or intuit the existence of abstracta in the appropriate circumstances. But this route fares no better. The chief difficulty for this approach, according to Alvin Goldman and Joel Pust, is that intuition doesn’t seem capable of delivering the information needed, and so can’t be the source of our knowledge of abstracta. They ask
Is there any reason to suppose that intuitions could be reliable indicators of a universal’s positive and negative instances (even under favorable circumstances)? The problem is the apparent “distance” and “remoteness” between intuitions, which are dated mental states, and a nonphysical, extra-mental, extra-temporal entity. How could the former be reliable indicators of the properties of the latter?26
It will have to take more than intuition to cross the epistemological gap between us and abstracta. To make matters worse, no other candidates appear even remotely plausible. Thus Moreland’s comment that “[such] an attempt to account for knowledge of abstract objects will have to be given in terms of some mysterious, even mystical, aphysical ‘grasping connection,’” which is philosophically repugnant, especially for a contender for (N) (no souls, no spooks, no entelechies, no gods).27 To this effect others have mentioned a sort of “argument from queerness” against platonism. Platonic entities are so foreign to and different from everything else in a naturalist ontology that special pleading against non-reductive theories threatens. In an oft-quoted passage, the late atheist J. L. Mackie wrote:
If there were [abstract objects], then they would be entities or qualities or relations of a very strange sort, utterly different from anything else in the universe. Correspondingly, if we were aware of them, it would have to be by some special faculty of…perception or intuition, utterly different from our ordinary ways of knowing everything else.28
Platonist Theories of Free-Floating Immaterial Objects Knowable Only Via Mystical Experience Prove Adverse and Excessive To (N).
Moreover, if there are sufficiently strong arguments in favor of conceptualism, they could count as evidence against both (1a) and (1b). Unfortunately most philosophers see no arguments here strong enough to sufficiently eliminate an alternative view such as platonism. However, this point is not entirely lost for at least two reasons. First, it can be argued that conceptualism has more scope than (1a) and (1b), as it nicely accounts for precisely the desiderata they are at odds with (more on this later). Second, it is the testimony of some philosophers that conceptualism has more intuitive support than its alternatives, or at least enough to make it rationally acceptable in the absence of compelling pros and cons.29 I’ve found this intuition shared by laypeople also. In my discussions about the ontological status of abstract objects with people wholly unfamiliar with philosophical parlance, I often receive the eager opinion that “they’re like, in your brain or mind” before I even mention conceptualism as an option. One instance was even with a Jr. High student! Plantinga gives a similar report: “It also seems plausible to think of numbers as dependent upon or even constituted by intellectual activity; indeed, students always seem to think of them as ‘ideas’ or ‘concepts’, as dependent, somehow, upon our intellectual activity.”30 Admittedly, this doesn’t prove that conceptualism is true, but I certainly think it is a consideration in its favor and hence out of favor with (1a) and (1b).
Furthermore, in a situation like this we might be able to justify our inference (to conceptualism) by appealing to the Principle of Credulity expounded by Richard Swinburne: certain beliefs with which agents find themselves are—in the absence of counter evidence—probably true; the mere fact that you have a belief is grounds for believing it.31 These points notwithstanding, there just might be a positive argument for conceptualism after all. Recent thought on the nature of intentionality may provide us one. Intentionality is the property of being of or about something (of-ness or about-ness). Many abstracta seem to be characterized by intentionality. For example, thoughts, as propositions, are of and about things. Possible worlds and (and perhaps states of affairs) consist of and are about descriptions of reality. The curious thing about intentionality is that it is characteristic exclusively of mental phenomena, so much so that it has been dubbed ‘the mark of the mental.’32 The implication is obvious: if intentionality is characteristic exclusively of mental phenomena and some abstracta have intentionality, then by modus ponens abstracta are characteristic of mental phenomena. This argument has the potential to be developed in much more detail (and, to some extent, already has33). What has been mentioned so far suffices to show how (2) could be met. From (1) and (2) it follows that
(3) Abstract objects exist and are mental concepts
So far any philosopher could accept our running argument regardless of whether God plays a role in their metaphysic. That is not to say, however, that (3) is metaphysically-neutral. Ardent naturalist David Armstrong, a main voice in the realism/anti-realism debate, is not dull to this point:
I suppose that if the principles involved [in analyzing and explaining reality] were completely different from the current principles of physics, in particular if they involved appeal to mental entities, such as purposes, we might then count the analysis as a falsification of Naturalism.34
Atheist philosopher Kai Neilson agrees:
Naturalism denies that there are any spiritual or supernatural realities. There are, that is, no purely mental substances and there are no supernatural realities transcendent to the world; or at least we have no sound grounds for believing that there are such realities for perhaps even for believing that there could be such realities. It is the view that anything that exists is ultimately made up of physical components. … There are no purely mental realities in a naturalistic account of the world.35
Armstrong’s and Neilson’s characterization of naturalism is clearly incompatible with (3). The defender of (N), on the other hand, can yet consistently maintain (3), though not without considerable tension (for reasons outlined above). But here the logical consistency between (N) and (3) ends by adding the explicitly theistic premise
(4) If abstract objects exist and are mental concepts, they must be concepts of a necessary, omniscient mind
What can we adduce for (4)? Why do we need a necessary, omniscient mind? The school of thought known for grounding abstracta in human minds is known as psychologism, traditionally associated with John Stuart Mill’s System of Logic. Psychologism, however, infamously stands as one of the most thoroughly refuted views in this field, beginning with the analyses of F. G. Frege and Edmund Husserl.36 Among phychologism’s insufferable difficulties is its inability to account for the necessity and sheer volume of abstratca. It would do precious little to explain the necessity of one entity in terms of another entity that is itself not necessary. Human minds are wholly inadequate for the task. Say some abstract object O is the concept of some human mind at time t1. Surely there must have been times ti–tn< t1 such that there were no human minds that had O as a concept. It would be less congenial to say that at t1 O came into existence than to say O must have existed as a concept of a necessary mind during ti–tn. This argument runs backwards as well: given their voluminous and complex nature, there must be abstract objects that have not yet nor will ever be concepts in human minds.
Moreover, it’s a bit too optimistic (even for the philosopher!) to think, as Abraham Robinson remarked, that we can “comprehend the incomprehensible,” such as the hierarchal order of sets or some immensely complex mathematical formula. The existence and truth of abstracta such as these is obviously independent of our cognizing them. Quentin Smith, for example, imagines an infinitely complex conjunct of all true propositions as itself a single proposition. Such a proposition, Smith argues, could only be the accusative of an omniscient mind. In addition to propositions, Plantinga suggests reasons for thinking the same is true of sets, numbers, and properties as well.37 Alternatively, It is not hard to see how the theist’s version of conceptualism would be immune to the standard objections to psychologism. It is also not hard to see how the theist’s version of conceptualism would be a hand-in-glove fit with precisely the desiderata these other theories of abstracta are at odds with: knowability despite causal impotence, unextendedness, and necessity. But if (3) is true and other theories like psychologism that try to “drag everything down to earth out of heaven and the unseen” cannot bear the argument’s weight, then plausibly
(5) Therefore, a necessary, omniscient mind exists
which turn entails ~(N). But the theist shouldn’t be too hasty. It remains to be shown that the conclusion is coherent. If demonstrated that (4), for example, is in some way incoherent, the conceptualist argument would be unsound (here one thinks of Morris and Menzel’s theistic activism). No doubt the theist will also have to spell out in much more detail the version of conceptualism needed. For example, even if the theist establishes conceptualism, it still needs to be said how abstract objects become concepts in human hinds. One can predict the Christian’s appeal to the imago Dei within us, the blueprint of which includes knowledge of abstracta. Moreover, the inherent fallibility and error in our best efforts at reasoning with abstracta would not be surprising given the cognitive consequences of sin. Robert Adams entertains these possibilities:
If God of his very nature knows the necessary truths, and if he has created us, he could have constructed us in such a way that we would at least commonly recognize necessary truths as necessary. In this way there would be a causal connection between what is necessarily true about real objects and our believing it to be necessarily true about them. It would not be an incredible accident or an inexplicable mystery that our beliefs agreed with the objects in this.38
Forthcoming efforts on these and similar issues are already underway.39 At any rate, the existence and nature of abstract objects might prove to be either at odds or flatly inconsistence with naturalism whereas no such difficulties loom for theism. Indeed, abstract objects may provide the tools for a promising theistic argument. I therefore agree with eminent atheist philosopher Quentin Smith that “the conceptualist argument…may be interpreted as contributing to establishing the rational acceptability of theism.”
1. Thomas Morris and Christopher Menzel, “Absolute Creation” American Philosophical Quarterly 23 (1986): 353-362. Several other relevant pieces are Menzel’s “Theism, Platonism and the Metaphysics of Mathematics” Faith and Philosophy 4/4 (1987): 365-382; “God and Mathematical Objects” in eds. Russell W. Howell and W. James Bradely, Mathematics in a Postmodern Age (Eerdmans, 2001). Brian Leftow, “A Leibnizian Cosmological Argument” Philosophical Studies 57 (1989): 135-155; “Is God an Abstract Object?” Noûs 24 (1990): 591-598; “God and Abstract Entities” Faith and Philosophy 7/2 (1990): 193-217. Scott A. Davison, “Could Abstract Objects Depend Upon God?” Religious Studies 27 (1991): 485-497. Matthew Davidson, “A Demonstration Against Theistic Activism” Religious Studies 35 (1999): 277-290. Michael Bergmann and Jeffrey Brower, “A Theistic Argument Against Platonism(and in Support of Truthmakers and Divine Simplicity)” Oxford Studies in Metaphysics 2 (2006): 357-386. No doubt a primary influence behind all of this was Platninga’s 1980 Aquinas Lecture, “Does God Have A Nature?”.
2. Alvin Plantinga, Warrant and Proper Function (Oxford, 1993), 121 fn. 25. See also his “How to be an Anti-Realist” APA Proceedings and Addresses (1982), 47-70, where a version of conceptualism is defended in some detail. Realism is often called platonism and anti-realism, nominalism. But some realist theories demur with platonism and some nominalist theories are not quite anti-realist. Moreover, I will use the terms “entity” and “object” interchangeably, though mindful of E. J. Lowe’s useful distinction in “The Metaphysics of Abstract Objects” Journal of Philosophy 92/10 (1995): 509-524. Furthermore, I leave the notion of ‘concept’ undefined, though aware of the desperate need of clarifying its relation to abstracta.
3. See Ted Sider’s impressive though incomplete “Bibliography on Abstract Entities”.
4. Gideon Rosen, “Abstract Objects” in Stanford Encyclopedia of Philosophy. See also the entry “God and Other Necessary Beings” by Matthew Davidson.
5. “Does God Exist?” a debate between William Lane Craig and Corey G. Washington on February 9, 1995 at the University of Washington.
6. . See Neil Tennant, “On the Necessary Existence of Numbers” Noûs (1997): 321. George Bealer, “Universals” Journal of Philosophy 90/1 (1993): 5-32. Sets with contingent members seem to be an exception to the criterion of necessary existence.
7. On the notion of a null-world and God’s relationship to propositions, see the excellent exchange between Richard Davis and Brian Leftow in Religious Studies 42 (2006). See especially Davis’s article, “God and Counterpossibles”, 371—391.
8. Thomas V. Morris, Our Idea of God (InterVarsity, 1991), 109
9. Ibid, 110.
10. J. P. Moreland, “Naturalism and Libertarian Agency” Philosophy and Theology 10/2 (1997): 353.
11. Howard Robinson, Matter and Sense (Cambridge, 1982), 50. On the incompatibility of realism and naturalism, see J. P. Moreland, “Naturalism and the Ontological Status of Properties”, in ed. William Lane Craig and J. P. Moreland, Naturalism: A Critical Analysis (Routledge, 2000), 67-109.
12. See his Review of: Naturalism: A Critical Analysis. I specifically use Oppy’s definition to preempt the sort of hand waving he does in his review, dismissing the entire book on the basis that the contributors didn’t define naturalism to his liking!
13. W. V. Quine, “On What There Is”, in ed. Stephen Hales, Metaphysics: Contemporary Readings (Wadsworth, 1999), 212. What Quine calls realism I’m calling platonism.
14. See Torsten Wilholt, “Think about the Consequences! Nominalism and the Argument from the Philosophy of Logic” Dialectica 60/2 (2006): 115–133.
15. Surprisingly Josh Parsons does not consider this in his “There is no ‘Truthmaker’ Argument against Nominalism” Australasian Journal of Philosophy 77 (1999): 325-334.
16. Alex Orenstein, W. V. Quine (Princeton, 2002), 1; 55-56. Contrast Quine’s earlier views expressed in his paper “Steps Toward a Constructive Nominalism”, coauthored with Nelson Goodman: “We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.—exist in space-time; but we mean more than this. We renounce them altogether…Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.” Idem., 55.
17. On Quine’s (and Putnam’s) indispensability argument see Mark Colyvan, “In Defence of Indispensability” Philosophia Mathematica 6 (1998): 39-62. I reword their argument to broaden its scope. To defend the argument so-worded would call for precision on what exactly “experiential framework” means and why we must have ontological commitment to those entities indispensable to it. See also Mark Colyvan’s Stanford Encyclopedia of Philosophy’s entry, “Indispensability Arguments in the Philosophy of Mathematics”.
18. On fictionalism see Hartry Field, Realism, Mathematics and Modality (Blackwell, 1989). Mark Balaguer, Platonism and Anti-Platonism in Mathematics (Oxford, 1998); “A Theory of Mathematical Correctness and Mathematical Truth” Pacific Philosophical Quarterly 82 (2001): 87-114. Jody Azzouni, Deflating Existential Consequence: A Case for Nominalism (Oxford, 2004).
19. Richard Swinburne takes this route in The Christian God (Oxford, 1994), 96-116. On this topic in general see Paul Copan and William Lane Craig’s excellent chapter “Creatio ex Nihilo and Abstract Objects” in Creation out of Nothing (Baker, 2004), 167-195. See also Craig’s concise summary of the fictionalist position in his “Current Work on God and Abstract Objects”.
20. Tennant, “On the Necessary Existence of Numbers”, idem. A standard fictionalist rendition would be(#) For any sort of entity F, There are n Fs iff according to the numbers fiction, The number of Fs = nBut this is precisely the sort of language that Tennet claims incurs ontological commitments. For more on this point against fictionalism, see Daniel Nolan and J. O’Leary-Hawthorne, “Reflexive Fictionalisms” Analysis 56 (1996): 26-32. See also Matti Eklund’s, “Fictionalism” in the Stanford Encyclopedia of Philosophy.
21. Tennant, “On the Necessary Existence of Numbers”, 322.
22. Argument adapted from John Baggaley, “The Ontological Argument for the Existence of God” 25
23. Wes Morriston, “Craig on the Actual Infinite” Religious Studies 38 (2002): 151.
24. Quoted in Reuben Hersh, What Is Mathematics, Really? (Oxford, 1999), 42. For more on this point against platonism see William Lane Craig, The Kalam Cosmological Argument (Wipf and Stock, 1979. ed. 2000), 88-94. However, this modus tollens is another’s modus ponens: some claim the existence of abstracta is proof positive of an actual infinite. On this see J. P. Moreland, “A Response to a Platonistic and Set-theoretic Objection to the Kalam Cosmological Argument” Religious Studies 39 (2004): 373-390.
25. For powerful presentations of this argument, see Paul Benacerraf’s “What Numbers Could Not Be” Philosophical Review 74 (1965): 47-73; “Mathematical Truth” Journal of Philosophy 70 (1973): 661-79. More recent and exhaustive is Colin Cheyne’s Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism (Springer, 2001). Benacerraf et al. have also advanced what is called the uniqueness objection against platonism. Frankly I am not able to make much sense out of that argument.
26. Alvin Goldman and Joel Pust, “Philosophical Theory and Intuitional Evidence” in eds. Michael R. DePaul and William Ramsy, Rethinking Intuition: The Psychology of Intuition and Its Role in Philosophical Inquiry (Rowman & Littlefield, 1998), 185.
27. J. P. Moreland, Universals (McGill-Queen’s, 2001. ed. 2005), 121. For more on the incompatibility of platonism and naturalism, see Jarrold J. Katz, Realistic Rationalism (MIT, 1998), 25-83. Laurence Bonjour, In Defense of Pure Reason (Cambridge, 1998). One important rejoinder to the epistemological argument I did not mention is Mark Balaguer’s. Balaguer argued that if every possible entity e in some category C existed, then we can have knowledge of extra-mental entities without a causal relation simply by imagining or “dreaming up” any e from C. Given that every e from C exists, we must be cognizing at least some actual entities of which C is composed. Now if we assume that every possible abstract object that could exist does exist, then it follows that the ones we believe exist do exist. See Mark Balaguer, Platonism and Anti-Platonism in Mathematics, 49. Three responses are in order. First, it is interesting to note Tennet’s argument for the primacy of zero, if successful, rebuts Balaguer’s argument because it demonstrates that it is false that every possible mathematical entity exists. Second, it is doubtful that the act of “dreaming up” or imagining some e, even if that e existed, would be sufficient for genuine knowledge of e to obtain. And third, even if we granted that every e that could exist does exist and that our act of imagining some e is sufficient for knowledge of e, we would still run into the problem of the actual infinite. For if every e that could exist does exist, there must be an actual infinite number of es. For a more detailed criticism of Balaguer’s argument, see Colin Cheyne, “Problems with Profligate Platonism” Philosophia Mathematica 7 (1999): 164-177.
28. J. L. Mackie, Ethics: Inventing Right and Wrong (Penguin, 1977), 38. Mackie is speaking specifically of objective values here, but such would themselves be abstract objects.
29. See Alvin Plantinga, Does God Have A Nature? (Marquette, 2003), 127-140; “How to be an Anti-Realist”, 67-68. Quentin Smith, “The Conceptualist Argument for God’s Existence” Faith and Philosophy 11 (1984): 45-49.
30. Alvin Plantinga, “Two Dozen (or so) Theistic Arguments,” as printed in ed. Deane-Peter Baker, Alvin Plantinga (Cambridge, 2007), 213.
31. See his The Existence of God (Oxford, 2nd ed. 2000), 303-315. See also Alvin Goldman and Joel Pust, “Philosophical Theory and Intuitional Evidence”, 181.
32. Tim Crane, “Intentionality as the Mark of the Mental” in ed. Anthony O’Hear, Current Issues in Philosophy of Mind (Cambridge, 1998), 229-252.
33. Richard Davis is pioneering work in this area. See his recent “God and Modal Concretism” Philosophia Christi 10/1 (2008): 57-74.
34. David M. Armstrong, “Naturalism, Materialism, and First Philosophy” Philosophia 8 (1978): 262.
35. Kai Nielson, “Naturalistic Explanations of Theistic Belief” in eds. Philip Quinn and Charles Taliaferro, A Companion to Philosophy of Religion (Blackwell, 1997), 402.
36. G. Frege, The Foundations of Arithmetic, trans. J. L. Austin (Northwestern, 1980). Edmund Husserl, Logical Investigations (Routledge, 2001).
37. Quentin Smith, “The Conceptualist Argument”, 40. Alvin Platninga, “Two Dozen (or so) Theistic Arguments”, section I arguments (a)-(c). Op cit., 210-213.
38. Robert M. Adams, The Virtue of Faith (Oxford, 1987), 218.
39. See especially Greg Welty’s, An Examination of Theistic Conceptual Realism as an Alternative to Theistic Activism (Oxford, 2000). John Byl, “Theism and Mathematical Realism” Proceedings of the Association of Christians in the Mathematical Sciences (2001): 33-48. Vern S. Poythress, “Reforming Ontology and Logic in the Light of the Trinity: An Application of Van Til’s Idea of Analogy” Westminster Theological Journal 57/1 (1995): 187-219; “Creation and Mathematics; Or What Does God Have To Do With Numbers?” The Journal of Christian Reconstruction 1/1 (1974): 128-140.
40. Quentin Smith, “The Conceptualist Argument”, 48.