Abstract
We agree with Hilbert’s assessment that the concept of ‘infinite’ stands in need of clarification – however, our proposed ‘solution’ is almost diammetrically opposed to that of Hilbert.
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Notes
- 1.
A nice example here is provided by that would-be ‘paragon’ of atheism, A.J. Ayer: “A being whose intellect was infinitely powerful would take no interest in logic and mathematics. For he would be able to see at a glance everything that his definitions implied, and, accordingly, could never learn anything from logical inference which he was not fully conscious of already. But our intellects are not of this order.” (1932, p.82) Compare also here Read’s argument against Michael Dummett, who charmingly admits the theological root of his philosophising around finite and infinite: see Read 2008a.
- 2.
And our scepticism has of course Wittgensteinian as well as Nietzschean grounds: “It isn’t just impossible ‘for us men’ to run through the natural numbers one by one; it’s impossible, it means nothing.” (Wittgenstein, PR, p.146). For amplification, see for instance Read’s “What does ‘signify’ signify?”, Read 2001a and resolute readings of Wittgenstein’s philosophy.
- 3.
This is a justly famous example from Wittgenstein; see e.g. PI para.s 350–1.
- 4.
As we shall briefly surmise in the Conclusion, this is arguably why Buddhist thinking at its best — in some Zen, and in Nāgārjuna, for instance—does not trust any assertoric let alone theoretical thought.
- 5.
See below for discussion of this in specific connection with a famous ‘locale’ for the discussion: Wittgenstein’s Tractatus.
- 6.
Cf. Coulter 1991
- 7.
See for instance the essays collected in Part II of The New Wittgenstein (Crary and Read, 2000), especially that by Conant (and excluding that by Hacker, a leading exponent of the ‘ineffabilist’ reading of Wittgenstein that we are here challenging as inadequately radical, inadequately characterising the difference between saying and showing, by means of making the latter look like just a deviant or mysterious version of the former). See also the arguments made by Read and Deans (2000). These papers make clear that we resolutists are the true friends of showing. It is the ineffabilists who fail to take ‘showing’ sufficiently seriously – for they covertly render it as just another form of saying.
- 8.
For discussion, see Read’s work critiquing Louis Sass, e.g. his “On approaching schizophrenia through Wittgenstein (2001b)”, and Sass’s “Heidegger, schizophrenia and the ontological difference”(1992), 109–132. Schizophrenics make the character of ‘the ontological difference’ abundantly clear, by virtue of their so palpably mistaking ontological matters for ontic ones, whereas philosophers (such as Descartes) make the same error, but so much less obviously.
- 9.
For discussion, see Wittgenstein’s Lectures and Conversations on Aesthetics, Psychology and Religion, especially Lecture I of the Religion lectures.
- 10.
For explication, see Hutchinson, Read & Sharrock 2008.
- 11.
See Frege’s great critique of Kerry, and discussions thereof by Conant, Read, etc.. When one tries to speak of the concept horse, one misses one’s target. When one tries to speak of infinity, one misses one’s target. Whenever we imagine infinity, we miss it. For ‘infinity’ is ontological, not ontic.
- 12.
It is perhaps worth noting here that some recent authors, most notably radically therapeutic Wittgensteinians Gordon Baker and Katharine Morris, have argued that the point made here is true only of ‘Cartesianism’, not of Descartes’s own views. This interesting exegetical question is beyond the scope of our paper to investigate.
- 13.
- 14.
Compare also PI 304.
- 15.
This sense of mind, as too deeply different from the corporeal to be adequately represented by the term, “incorporeal process”, combined with the sense of mind found in thinking of the mind as through and through embodied, together make up our concept of mind, and indeed of consciousness.
- 16.
See Palmer’s Concept and object 1988, and the essays by Conant and Witherspoon in The New Wittgenstein (Crary & Read 2000) and in Wittgenstein in America (McCarthy & Stidd 2001) on Ryle’s ‘Carnapian’ tendency to use the concept of ‘category mistake’ in ineffaciously and unpersuasively seeking to present perspicuously ontological differences. The term ‘category mistake’ tends to lead one systematically to under-estimate the difference between the two terms/‘things’ one is comparing, the categories being ‘mistaken’ for each other. ‘Mythic categorial nonsense’ might be a more apposite term than ‘category mistake’ for averting this risk. But any term alone is not enough; the actual work of averting philosophical delusion in oneself and in others always remains to be done, after one has picked the terms least likely in any given context to confuse or delude one in the course of that therapeutic work. Here is an example of where the concept of a “category mistake” or “conceptual transgression” is leading a would-be Wittgensteinian philosopher astray: “the problem rests on the illicit use of finite-bound concepts for infinite series” (Shanker 1987, p.183).
- 17.
- 18.
Compare PI 36.
- 19.
It is worth noting that, actually, this infinitude of sentences only even appears to be generated if we stick to a syntactic (or possibly a pragmatic) version of ‘sentence’. If, by contrast, one defines ‘sentence’ semantically, as something that actually means something (rather than just a string of nothing), then eventually one comes back to sentences one already encountered.
- 20.
Compare also this remark: “If we pay close attention, we find that the literature of mathematics is replete with absurdities and inanities, which can usually be blamed on the infinite” (Hilbert 1925, p.370). Hilbert’s approach, different from Watson’s, nevertheless uses much the same idea.
- 21.
In connection with the traditional concepts of ‘potential’ and ‘actual’ infinity, we might express our point as follows: The standard view is that ‘actual’ infinity would be (a real) infinity. But an ‘actual’ infinity is a would-be thing. And infinity is entirely mis-identified as some kind of a thing. We would thus be inclined to reverse the polarity; a broadly Wittgensteinian approach should see the notion of potential infinity as coming closer to what one wanted to index when one spoke of infinity. Infinity is, as it were: there can always (potentially) be more.
This could be seen as a radicalisation of a thought already present in Aristotle, and obscured in the prose of Hilbert, and at times of Cantor.
- 22.
And some people really do think this: “Infinite totalities cannot depend on us for their existence; if they exist, they must exist independently of our states of mind, of our thoughts, and of the course of our experience. [...] mathematicians seem able to direct their minds toward, to think cogently about and accumulate compelling truths about, infinite mathematical entities. [...] there seems to be a parity of the following sort between the existence of infinite mathematical entities and states of affairs in the world: they exist independently of us, and this means at least that thoughts ‘about them’ are true or false independently of our states of mind and of what we happen to know” (Tragesser 1984, p.38).
- 23.
For fleshing out of these metaphors, see Hutchinson & Read 2008.
- 24.
This has to do with the false dichotomy in Latour and Woolgar between ‘Science is superhuman’ and ‘Science is banal’. Kuhn’s account in this ballpark is more helpful: see Sharrock and Read’s Kuhn (2002) for discussion of the importance of science being hard, and involving lots of work, and of the neglect of this important fact in most philosophy of science, except in Kuhn and some of those he has influenced.
- 25.
On the separation of maths and philosophy, see our paper on Godel, forthcoming.
- 26.
Siderits & O’Brien 1976. This paper is of particular interest (in the context of the festschrift for Siderits which the present paper is part) because of its being an early piece of Siderits’s that doesn’t (unlike well-known instances of his more recent thinking) mention Wittgenstein but that is specifically about the topic of our paper (in a manner that we find obviously Wittgensteinian): namely, infinity (as a topic within which to clarify the nature of philosophy).
- 27.
The fuller quotation, from p.288 of Siderits and O’Brien (1976), reads: “Nāgārjuna’s chief task in MMK is to provide a philosophical rationale for the notion of sunyataor “emptiness,” which is the key term in the Prajfiparamita Sutras, the earliest Mahayana literature. …To this end Nāgārjuna constructs a dialectic which he considers capable of reducing the metaphysical theories of his opponents (chiefly Sarvastivada, Sramkhya,and Nyaya) either to contradiction or to a conclusion which is unacceptable to the opponent. Unlike Zeno, however, Nāgārjuna is not refuting the theories of his opponents simply as a negative proof of his own thesis: Nāgārjuna has no thesis to defend-at least not at the object-level of analysis where metaphysical theories compete with one another. Instead his dialectic constitutes a meta-level critique of all the metaphysical theses expounded by his contemporaries.” (Italics ours.) The anticipation of Wittgenstein methodologically here is striking. Cf. also p.297 of Siderits and O’Brien.
- 28.
Cf. also p.298 of Siderits and O’Brien; their exegesis according to which for Nāgārjuna “metaphysics is a fundamentally misguided under- taking” is strikingly anticipative of Wittgenstein’s approach. (For a thorough comparison of Buddhist methodology in philosophy to Wittgensteinian methodology, see Read 2012b, Chap. 8.)
Abbreviations
- LFM :
-
Lectures on the Foundations of Mathematics, Ludwig Wittgenstein
- PG :
-
Philosophical Grammar, Ludwig Wittgenstein
- PI :
-
Philosophical Investigations, Ludwig Wittgenstein
- PR :
-
Philosophical Remarks, Ludwig Wittgenstein
- ROM II :
-
Remarks on the Foundations of Mathematics volume 2, Ludwig Wittgenstein
- WVC :
-
Wittgenstein and the Vienna Circle
References
Ayer, A.J. 1932. Language, Truth and Logic. London: Penguin.
Collins, H.M. 1985. Changing Order: Replication and Induction in Scientific Practice. London Beverly Hills: Sage Publications.
Coulter, J. 1991. Is the `New Sentence Problem’ a Genuine Problem? Theory & Psychology 1 (3): 317–336.
Crary, A., and R. Read, eds. 2000. The New Wittgenstein. London: Routledge.
Davidson, D. 1967. Truth and Meaning. Synthese 17: 304–323.
Hilbert, D. 1925. On the Infinite.
Hutchinson, P., and R. Read. 2008. Toward a Perspicuous Presentation of “Perspicuous Presentation”. Philosophical Investigations 31 (2): 141–160.
Hutchinson, P., R. Read, and W. Sharrock. 2008. There Is No Such Thing as a Social Science: In Defence of Peter Winch. Aldershot: Ashgate.
Knuth, D.E. 1976. Mathematics and Computer Science: Coping with Finiteness. Science 194 (4271): 1235–1242.
Latour, B., and S. Woolgar. 1979. Laboratory Life: The Construction of Scientific Facts. Princeton: Princeton University Press.
McCarthy, T., and S. Stidd, eds. 2001. Wittgenstein in America. Oxford: Oxford University Press.
Niles, I. 1992. Wittgenstein and Infinite Linguistic Competence. Midwest Studies in Philosophy 17 (1): 193–213.
Palmer, A. 1988. Concept and Object: The Unity of the Proposition in Logic and Psychology. London/New York: Routledge.
Quine, W.V.O. 1960. World and Object. Cambridge, MA: The MIT Press.
Read, R. 2001a. What Does ‘Signify’ Signify?: A Response to Gillett. Philosophical Psychology 14 (4): 499–514.
———. 2001b. On Approaching Schizophrenia Through Wittgenstein. Philosophical Psychology 14 (4): 449–475.
———. 2002. Is ‘What Is time?’ a Good Question to Ask? Philosophy 77 (2): 193–209.
———. 2008a. Applying Wittgenstein. London/New York: Continuum.
———. 2008b. The ‘Hard’ Problem of Consciousness is Continually Reproduced and Made Harder by All Attempts to Solve It. Theory, Culture & Society 25 (2): 51–86.
———. 2012. A Wittgensteinian Way with Paradoxes. Plymouth: Lexington Books.
Read, R., and R. Deans. 2003. “Nothing is Shown”: A ‘Resolute’ Response to Mounce, Emiliani, Koethe and Vilhauer. Philosophical Investigations 26 (3): 239–268.
Ryle, G. 1949. The Concept of Mind. Chicago: University of Chicago Press.
Sass, L. 1992. Heidegger, Schizophrenia and the Ontological Difference. Philosophical Psychology 5 (2): 109–132.
Shanker, S. 1987. Wittgenstein and the Turning Point in the Philosophy of Mathematics. Albany: State of New York Press.
Sharrock, W., and R. Read. 2002. Kuhn. Oxford: Polity.
Siderits, M., and J.D. O’Brien. 1976. Zeno and Nāgārjuna on Motion. Philosophy East and West 26 (3): 281–299.
Tragesser, R.S. 1984. Husserl and Realism in Logic and Mathematics. Cambridge: Cambridge University Press.
Watson, A.G.D. 1938. Mathematics and Its Foundations. Mind 47 (188): 440–451.
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Read, R., Greiffenhagen, C. (2023). Finite and Infinite: On Not Making ‘Them’ Different Enough. In: Coseru, C. (eds) Reasons and Empty Persons: Mind, Metaphysics, and Morality. Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-031-13995-6_16
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