by Dave Maier
This post started out as one about the mathematical question of whether 1 = 0.9999…, an issue which confuses a lot of people. This confusion shouldn’t be surprising, as it involves infinity; and if you’re not confused about infinity, then you probably don’t understand it. Unfortunately what I had to say about it, at least in the strictly mathematical context, has been said fairly well already by many others, which shouldn’t be surprising either, as I am not a mathematician. (I should have Googled it first and saved myself some time.)
So rather than going through all that stuff again (although we will see some of it soon enough), let me say here at the beginning why it was sticking in my mind in the first place. This will introduce some dramatic tension into an otherwise boring post, as the reader wonders what the heck these things have to do with each other. Like I said: another shaggy dog story to start the new year. (The pictured animal is to be found on the Wikipedia entry for “shaggy dog story” and is identified there as “the archetypical subject of long-winded, pointless stories”.)
So then. I just began reading Charles Taylor’s new book The Language Animal (seriously, I’m on page 6). Taylor has been writing about these things for many years, so his general views are already familiar, but apparently he has a bit more to tell us. His main concern, as he tells us on page ix, is the same as always: to argue that our linguistic capacity is “more multiform than has usually been supposed, [in that] it includes capacities for meaning creation which go far beyond that of encoding and communicating information, which is too often taken as its central form.” In particular, while Taylor allows that contemporary analytic philosophy of language is much more sophisticated than in early modernity, when rationalism and empiricism were the main players on the philosophical stage, “certain […] key assumptions” of that era “have survived into analytic post-Fregean philosophy”.
The reason we need a new book, it turns out, is that while progress has been made on this front, and the objectionable theory has at least been crushed into pieces, it has not yet been entirely pulverized.
An important task of The Language Animal is thus to “refute the remaining fragments” of this legacy, which he calls the “HLC” theory, named after its origins in the work of Hobbes, Locke, and Condillac. Taylor rejects this theory’s conception of linguistic meaning as “designative”, as well as, more generally, the form it takes, i.e. an “enframing” theory, which attempts to construct a conception of language out of pre-existing, non-linguistic elements (and indeed one can easily see how the two conceptions might reinforce each other).
Taylor’s favored replacement, as before, is the “HHH” theory, correspondingly named after its origins in the work of 18th- and 19th-century German philosophers [Johann Georg] Hamann, [Johann Gottfried] Herder, and [Wilhelm von] Humboldt. Taylor calls this a “constitutive” theory, which is the “antitype of the enframing sort”, and presents language as “not explicable within a framework picture of human life conceived without language” (p. 4). Here too, the theory’s form fits naturally with its content, for which Taylor in fact uses just about the same word (“constitutive-expressive”). On this view, language does not and, crucially, cannot simply denote objects in the world independent of ourselves (that is, in the way made mysterious by modern epistemology and metaphysics); it must also in some way help to constitute its object and at the same time (as I like to put it) express its user’s fundamental existential commitments.
Another of Taylor’s main influences has been the phenomenological tradition, especially the work of Maurice Merleau-Ponty. Here also we find a base from which Taylor attacks the residual fragments of the modern viewpoint, as found in even those contemporary analytic philosophers who have explicitly rejected that Cartesian inheritance. The Language Animal appeals mainly to the HHH crew, but I have also just started another book of Taylor’s, this one cowritten with Hubert Dreyfus, another big-name contemporary phenomenologist and long-time gadfly of Cartesian views (especially in cognitive science, as in his famous broadside against artificial intelligence, What Computers Can’t Do). Retrieving Realism (a title which throws down the gauntlet right from the get-go) seems to be mostly Dreyfus’s baby, reprising some of its material from an earlier volume on the debate about these matters between Dreyfus and John McDowell, but as we’ve already seen, it fits Taylor’s general picture as well. The basic idea is that the phenomenological turn to the primacy of what Dreyfus calls “engaged coping” is the only way to do fully what Davidson, Rorty, and McDowell do incompletely: to escape the broadly Cartesian picture of knowledge as a matter of inner representations mapping external reality.
We’ll get back into this some other time, after I’ve read more of The Language Animal (and maybe Retrieving Realism if I can stand it). My general attitude here is that Dreyfus, and Taylor too to at least some extent (although I’m willing to give The Language Animal a chance) is a) specifically, does not get what McDowell and/or Wittgenstein are doing; and b) more generally, does not see the potential for the modern philosophical tradition to transcend its roots in Descartes, Hobbes, et. al, and instead visits the sins of the fathers unto the last generation, demanding, as we have already seen, that even its “remaining fragments” be scattered to the four winds. Without calling this rhetoric “genocidal” or anything, I do think that this judgment is not simply extreme and unfair but also actually contrary to the anti-Cartesian insights appealed to by both the phenomenologists and the post-analytic philosophers (McDowell and his allies) they attack.
This ceaseless internal squabbling among those of us in the anti-Cartesian brigade gives renewed confidence to our enemies, who ignore such petty differences and brand us all indiscriminately as batty and irresponsible postmodern-relativist poltroons. I hold out some hope that the two sides (that is, post-analytics and phenomenologists) may each find our way to neutral territory, such as some manner of post-Gadamerian hermeneutics. But that, as I often say, is another story for another day.
Sooo, how does our long-forgotten mathematical issue fit into all this?
It does of course look funny at first when they tell us that the number 0.9999… does not simply approximate but is actually equal to 1. “But it never gets there!” we exclaim; and the unsophisticated objections to the idea generally explicitly or eventually take this form. More sophisticated objections are to the idea of infinity, or of degrees of infinity, or perhaps the real numbers themselves. This last might be surprising, as when they introduce the real numbers in math class, they wisely don’t get into the very real weirdness that the idea involves; this includes the complicity of the set-theoretical Axiom of Choice, which you can bet I will not be getting into today. In such cases, the debate quickly moves into the issue of whether the real numbers Actually Exist or whether I get to choose which numbers I may use for my own purposes – or if the real numbers are actually worth the trouble. Most people say they are, but that’s probably because they were hoping to have differentiable or at least continuous functions, which I must say are remarkably handy if you were thinking of using calculus.
But let’s get back to the unsophisticated objections, which are actually more to the point (such as it is). The aspect of same which interests me here is their apparent resistance to the idea that there could be two such different names for the same number. Of course we already know that the equation
7 + 7 = 14
says that the expressions on either side of the equals sign each denote the same number, denoted below in another way still:
o o o o o o o o o o o o o o
If we thought that the left side (“7 + 7”) is somehow cheating, since it’s really a sum of other numbers, and that “14” is the real name for that number, let’s note that even “14” is a sum (10 + 4), and that yet another name for this same number is “16”: that is, in base 8 (and so 8 + 6). I take it that no one wishes to argue that because of this the only real name for this number is “E” (the hexadecimal numeral), simply because it is a single digit and thus not a sum.
But what about
1 = 0.999999… ?
Here what is screwy about the right side is not simply that it is a sum, but that the sum is in some way supposed to be infinite. Here it is common to take this opportunity to manifest one’s confusion about the difference between the infinitely large, on the one hand, and the arbitrarily large but finite, on the other. This bit is the part which is better explained elsewhere, and I will not waste your time with it since it’s not really my point anyway, any more than the various clever but not strictly rigorous ways one might try to “prove” the identity, such as starting with “1/3 = 0.3333….” and multiplying both sides by 3. In any case, these arguments, if successful, will help convince objectors that the two expressions do indeed denote the same number.
Naturally I too wish to maintain the identity. But instead of doing so by arguing about the utility of the reals or the nature of infinity, let’s look at what it means for an expression to “denote” a number in the first place.
Let’s say, then, that I go ahead and concede the point which seems to be causing the trouble: when you start with 0 and append 9s one at a time, you never get a number equal to 1 even if you go on forever. That is, it’s not – retracting the apparently objectionable terminology – that you get a number actually equal to 1 when we add an infinite number of 9s, because (the objection goes) that can’t happen, even in theory. Fine, have it your way; but now let us ask which number is the lowest such number? Which number does 0.9999… [not equal, as you insist, but] approach?
Clearly that number is 1. That it can’t be anything short of 1 we can show without any theoretical baggage, as all we need is the idea of an arbitrarily long but finite string of 9s: whatever finitely long string of 9s makes up the supposed number less than 1 (i.e. which “0.999…” approaches), it can’t be long enough, as I can always give you another such number which is larger than that one and yet less than 1; therefore only 1 is large enough – and thus the smallest such number as well.
Again, talking this way seems at first to give up the idea that 0.999…. is equal to 1 in favor of the objectors’ idea that it simply approaches it. But even if that which is denoted by the expression 0.999… is not the mathematical object which is the number 1, that expression still (let's say) gestures at – and, importantly, does so uniquely – that same object.
This was another point where my original post came to seem redundant, as many online discussions of this issue pretty much do the same thing by appealing to the mathematical concept of limits. Crucially to my larger point, though, they usually do so to defend the equality after all, as opposed to giving it up to make a point.
But now this is where we reach the long-awaited punch line of our story. For in either case –- using the concept of limits to show that the same object is denoted by the two expressions after all, or instead giving up that idea and differentiating between strictly denoting an object and (merely) gesturing at it — we end up in pretty much the same place. That is, while it's true that what Taylor and Dreyfus reject as an unacceptably “designative” conception of language (here, mathematical symbols, but meaningful inscriptions just the same) can get us into conceptual difficulty, and substituting a more “engaged”, practical, less strictly formal relation of “gesturing” can help sidestep it, it seems false that such a strategy is necessarily different –- as it seems Dreyfus at least is claiming it is –– from simply being more careful about saying why our concepts denote (or “designate”) what they do.
Not only that, it seems that this is true even in the purely mathematical context, in the heart of the modern scientific-rational beast. Far from needing scorched-earth phenomenological demolition projects, even here in this most seemingly unavoidably Cartesian of conceptual spaces we have resources to hand which show a way out of that lamentable muddle — and thus potentially a fortiori in more congenially post-analytic contexts as well if we look. Anyway, that’s the thought. Woof!