Scott Aaronson's Rapid Proof of Godel's Incompleteness Theorems

I finally got around a few days ago to starting reading a book I've had for a while now, Scott Aaron's book of lectures on the foundations of mathematics, computational complexity, cryptography and quantum mechanics, Quantum Computing Since Democritus. I started with Chapter 3 "Gödel, Turing and Friends". One of the first things Aaronson does in this chapter (or lecture, since it’s basically a lecture) is prove Gödel's Incompleteness Theorems via a reproduction of Turing's proof of the impossibility of a solution to the halting problem. It's an extremely quick proof which I didn't know about and is cool because of how quick it is, so I'm going to reproduce it below for fun.

First, the proof that there's no program to solve the halting problem. This is actually a very simple reductio proof (for which Aaronson can’t take any credit). Suppose there is such a halting-problem-solving program $P$ (a program which can determine for any given, arbitrary program plus input to that program whether that arbitrary program will halt). Then, we can modify $P$ to produce a new program $P^{\prime}$ that does the following. Given another program $Q$ as input, $P^{\prime}$

(i)                  Runs forever if Q halts, or

(ii)                Halts if Q runs forever.

(How do we know we can modify $P$ to do this? Simply because getting a program to run forever or to stop are extremely easy problems, on Turing machines as when programming on a PC.)

$P^{\prime}$ must be able to take any arbitrary program as input, so let $Q$ be $P^{\prime}$. By the conditions above, $P^{\prime}$ will run forever if it halts, or halt if it runs forever. So $P^{\prime}$ cannot exist, and so neither can $P$, by implication (specifically, $P$ cannot exist by modus tollens because $Exists(P) \rightarrow Exists(P^{\prime})$ and $P^{\prime}$ does not exist).

So how does this relate to Gödel? First, we gotta remember what the first Incompleteness Theorem states. It says that, given a consistent, computable proof system there’s a true statement about the integers that can never be proved from that proof system. (What is a proof system, you ask? Axioms plus basic rules of inference). Let us use the reductio strategy once more and suppose that this were false – that is, that there existed a consistent, computable proof system $F$ from which any statement about integers could either be proved or disproved. Then, given a computer program:

 “We could simply search through every possible proof in $F$ until we found either a proof that the program halts or a proof that it doesn’t halt. This is possible because the statement that a particular computer program halts is ultimately just a statement about integers. But this would give us an algorithm to solve the halting problem, which we already know is impossible. Therefore, F can’t exist.” [22]

(The only element of rigour lacking from this proof is the unjustified claim that “the statement that a particular computer program halts is ultimately just a statement about integers”. Basically, what Aaronson is referring to is the fact that all computer programs –whether they involve sorting data structures, computing the shortest path in a $DAG$, factorising large numbers, configuring player models for a first-person computer game or whatever – are, ‘at bottom’, i.e. at the most primitive, machine-proximate level of encoding, just programs to solve basic maths problems with integers. This becomes clear in the part of your computer science degree when you analyse a C program in terms of Assembly Language, wherein you learn that all the higher-level syntax can be deconstructed in terms of a much simpler language which only has the ability to tell the computer to store numbers here, retrieve numbers from there, add or subtract two numbers, and move to this or that bit of code. Assembly language is “flipping bits” and “flipping bits” means altering binary sequences, i.e. essentially just changing the values of integers.

So all high-level programming is an elaborate re-description of – i.e. abstraction from – basic arithmetic problems (and the basic arithmetic problems are, in turn, a representation of the voltage patterns of the transistors). This, of course, makes it easier to write code for things that have nothing to do with basic arithmetic problems. Meanwhile, basic arithmetic problems are, so far as I know, all people bother to try to get Turing Machines to do, because anything more gets super fucked up (of course, the Church-Turing thesis implies that anything you can code in a high-level programming language could be redescribed in such a way that you could encode the exact same set of computations on a Turing Machine. The lambda calculus, which I learnt a few weeks ago, is perhaps the most beautiful abstract ‘model of computation’, because it is an algebraic mathematical system that allows one to fairly simply encode quite high-level programs, at least with the encoding for some basic functions (like the SUCC function or FIRST function) plus an encoding for a list. Thinking up Turing Machine designs for even simple problems seems to be way harder.))

Anyway, so that’s what that remark is about. It might seem still an unrigorous proof because of this remark, but whatever.

Now onto the second Incompleteness Theorem, which requires another detour via the halting problem.

Now we imagine another program $P$ which, given as input another program $Q$, tries to decide whether $Q$ halts by searching through every possible proof and disproof that $Q$ halts in some formal system $F$ (i.e. using the above strategy). Then, as before, suppose we modify $P$ to produce a new program $P^{\prime}$ that

(i)                  runs forever if $Q$ given its own code as input is proved to halt, or

(ii)                halts if Q given its own code as input is proved to run forever.

“Now suppose we feed $P^{\prime}$ its own code as input. Then we know that $P^{\prime}$ will run forever, without ever discovering a proof or disproof that it halts. For if $P^{\prime}$ finds a proof that it halts, then it will run forever, and if it finds a proof that it runs forever, then it will halt, which is a contradiction.” [22]

But this raises a problem, as Scott notes: “Why isn’t the above argument itself a proof that $P^{\prime}$ will run forever given its own code as input? And why won’t $P^{\prime}$ discover this proof that it runs forever – and therefore halt, and therefore run forever, and therefore halt, etc.?” Fortunately, he also provides us a solution, namely, that in “proving” that $P^{\prime}$ runs forever, we make use of the hidden assumption that the proof system $F$ is consistent.

And yet this raises a further puzzle, which leads us towards the Second Incompleteness Theorem. The above has the consequence that, “if $F$ could prove that $F$ was consistent, then $F$ could also prove that $P^{\prime}$ ran forever – thereby bringing back the above contradiction. The only possible conclusion is that if $F$ is consistent, then $F$ can’t prove its own consistency. This result is sometimes called Gödel’s Second Incompleteness Theorem.” [23]

Aaronson opines that the SIT establishes “what we maybe should have expected all along”. One way of putting the result, which he employs, is that if we want to prove that a theory $F$ is consistent, then “we can only do it within a more powerful theory” [23]. There is indeed something perfectly intuitive about this, to me at least.

Anyway, we’re done now. And so now you know how to prove the Incompleteness Theorems. Wasn’t that fun?

I'm back, baby! Did you miss me?

I'm back, so here's why I think I was gone in case you cared.. and also here are some typically rambling thoughts on matters related to the recent 'Sokal Squared' thing

Just had to update my blog for the sake of all the weird girls who stalk me. As to why I've had (by my standards) a massive break in posting, I can identify the following factors: 'external': getting into cricket again, being a little socially busier, spending a lot of time writing a long and well-researched essay on the future of the planet for a writing competition on which I'll soon get feedback (meaning I can post this essay here), and, a little more recently, obsessively playing the classical guitar, after not playing at all for nearly four years straight (and I mean obsessively); 'internal': being extremely happy and mentally relaxed (connected: being socially busier than I was), reaching a tipping point of general disgust with the amount of dogmatic and dimwitted intellectual pollution on the internet and not wanting to contribute to it with a non-excellent, non-cogent post. I've thought for a while I need to write something at some point on the early history of mathematics (in particular, how long it took 'us' to develop the notation that makes 'simple' maths so much easier) and what it tells one about the Flynn Effect, about the role of cognition-externalisation, the importance of language and notation to thought, and on the secret theory-ladenness of many things we forget are theory-laden. I think I will do this eventually. I also keep wondering if I should write reviews of things, like the books and movies I've enjoyed recently (or even not so recently), because that might be fun. The reason I didn't do that in the past with books is because I had this major pathological aversion to summarising things that I read and liked, because I realised that summary always costs nuance and information that the author included for a reason - usually, I feel like almost nothing is indispensable in books I like, so I prefer to quote chapters out of whole cloth. But maybe I should start reviewing. On the other hand, I have this disposition against which I think is a perverse manifestation of the 'sunk-cost' fallacy: my brain secretes this weird thought like, It would be wrong to start reviewing things now, because that's not your thing; you could have reviewed books and movies that you liked years ago but you didn't so you can't just go back on that now.

Anyway, I did have something non-narcissistic to say, induced by the "Sokal Squared" Hoax that just happened. I'm not really going to comment on the Hoax per se (on the ethics or intellectual 'validity' of it, or on the perpetrators (the truth is I don't actually have a strong opinion on, like, the ethics of the Hoax - I think Boghossian and Lindsay seem like massive wankers but whatever)), but I wanted to say something  on related matters.

Sturgeon's Law tells us that most of academia is total bullshit (what I mean is: that should be our default assumption). Most research, even in the harder sciences, goes absolutely nowhere, retreads old ground in a pointless way, or is unrigorous (I hesitate to write disjunctions like these, because one is wont to miss disjuncts (there are probably more failure modes)). Arguably, fundamental physics - that most reverent of scientific disciplines - has been little different from metaphysics for the last several decades (see Sabine Hossenfender's controversial new book (which I haven't actually read, I should admit, though I have read her blogposts, on her blogpost blog)). And, obviously, macroeconomics has some well-known issues with predicting shit, medical science has some serious issues with publication bias and corporate influence, and social scientists are, for the most part, fucking useless idiots who need to increase their n and stop p-hacking shit. (I could link to relevant articles but I'm happy to assume that the kind of people who read my blog are big intellectual playas who have already read such articles).
But in my humble opinion, if there's one kind of area of academia that's worse than all the others, it's specifically the unrigorous part of academia which is also completely obscurantist - the academia of long sentences and pollysyllabic verbiage (or, alternatively, as one can see, for example, in some parts of economics and probably some other disciplines, the academia of obscurantist mathematics - glyphs and runes included to dazzle rather than to give a rigorous structure to a problem). One commonly used shorthand for obscurantist academia is "Postmodern" academia. Whilst I don't really think it poses much of a problem for this term that Postmodern academics themselves think it's painfully simplistic and bigoted to use one term to encompass several decades of evolving scholarship (because I don't give a shit about obscurantist scholarship), I already implied in the preceding parenthetical interjection that there is one problem with using this term as a shorthand: namely, that obscurantist academia is bigger than the parts of academia where words/phrases like "Marx", "Foucault", "Derrida", "Barthes", "Kristeva", "Lacan", "Bourdieu", "Butler", "theory", "hermeneutics", "problematics", "hegemony", "towards a", "beyond the", "territorialisation", "post-structuralist", "meta-textual", and so on, are highly frequent, and an extraordinary degree of prolixity is commonplace (see Postmodernism Generator). 
To be sure, Postmodern academia is definitely a major bastion of such obscurantism and it's probably deserved some of the specific critiques it has gotten (like these two I have linked before; http://faculty.georgetown.edu/irvinem/theory/Nussbaum-Butler-Critique-NR-2-99.pdf, http://bactra.org/chomsky-on-postmodernism.html). Postmodern academia is the area to which the "X studies" fields (the main target of the Hoax) largely belong, along with English literature and (to a lesser extent) sociology and history (I am sure there are some gender, race-studies and English literature academics who are not so into abstruse verbiage and the French theoreticians who perfected it to an artform but prefer to write clearly (which does not necessarily mean their scholarship has merit, but I'm sure some of it is interesting and incisive and cogent and all those other nice things)). 
However, one issue with the critique of Postmodernism that fails to link Postmodern writing with obscurantism generally is that it gives Postmodern critiques an opening to say either "You just don't understand our technical language and theoretical constructs" or "You just don't like our political conclusions". This is shitty, because, in my opinion, the important thing to point out is that Postmodern obscurantism is not really any different, from an epistemological perspective, from any other kind of obscurantism, e.g. religious obscurantism, the 'deepities' of pop-philosophy and self-help books, bureaucratese, Fascist mysticism, or the obscurantism of so much of the 'canon of philosophy' going back centuries (the part that one of a crude bent of mind might classify as "counter-Enlightenment" (e.g. the whole German Idealism thing)). It's not different, from an epistemological perspective, because it's all equally vague, non-naturalistic BS, or, as I prefer to say, bad poetry. As Chomsky points out, the difference between mathematics being used properly as part of a rigorous scientific theory and polysyllables being used in a prolix sentence is that mathematics, used properly, can allow one to give a rigorous testable structure to theories and to find precise solutions to very difficult problems that one couldn't even approach without the language (think of the analytic power and beauty of a matrix (or just think of trying to do physics problems without maths lmao)). But when you decode a typical sentence of Latinate polysyllables in the works of Derrida, Lacan or Butler, it turns out either to be a thought expressible more simply or just pure gobbledegook. 
Why can't truth be found in long, turgid sentences populated with esoteric Latinate words? Well, for one thing, humans are really bad at even parsing long sentences from a memory point of view. (So, just as a kind of abstract theoretical point, supposing you did have something really deep to say, don't you think you'd want to make more of an effort?) For another thing, fuzzy words almost necessarily mean fuzzy thought and they definitely mean ambiguity; hence why a key development of the Enlightenment was this thing called a technical term (a term you explicitly define in the context of some kind of rigorous, self-contained usually mathematical framework). More to the point, I agree with the meta-philosophy and largely with the metaphysic explicated in Ladyman and Ross' magnum opus, Every Thing Must Go. I think that philosophers and theoreticians who don't practise metaphysics in the Quinean fashion, as simply a process of making rigorous the ontological commitments of our best scientific theories, are "neo-scholastics" talking nonsense for no humanly important end. Basically, the long and short of it is that science is hard, naturalistic metaphysics has to be a very humble enterprise which pays due fealty to science, and all scholarship which talks about matters of reality and existence without a serious connection to relevant scientific inquiry is BS.
Yes, ok, I admit it, I'm a "neo-positivist" (which is very different from "logical postivism", per se, because literally everyone agrees that that is a faulty doctrine for various, largely esoteric reasons I will not get into). Which, in truth, means I can't win with Pomo people, because if the other rebuttals don't stick, "Positivist" is the one they are bound to hurl. I mean, I should clarify that I don't actually like the label "neo-positivist", even though some people with similar views to mine call themselves that. What I prefer to call myself is a "neo-pragmatist structural realist", where "pragmatist" here refers to a pragmatist attitude to metaphysics, which says that we should avoid the idea that we can actually do metaphysics 'properly' and instead just accept that we should call real whatever is a projectible phenomenon or cluster of phenomena in the fully mind-independent world (a phenomenon whose postulation allows us to make systematic predictions that help us achieve goals in our navigation in the world as organisms), most of which phenomena just have to be 'read off' scientific theories, with fundamental physics taking priority simply because of its generality (see the book Every Thing Must Go(2007) for more). Basically, this means that I think a whole lot of analytic philosophy is useless, along with virtually all of Continental philosophy and X studies stuff. (Basically, what it means in practice is that when anyone ever tries to sound profound using fancy words, my reaction is "That's some nice poetry". Writing, to me, falls into one of the following categories: poetry, prose (as in prose fiction), science, failed science, or naturalistic philosophy.)
Now, I've been reading some history books lately, and I want to make clear: this is not to imply that, like, history is a waste of time if you're not going full Peter Turchin or some shit. Archival research and analysis would be necessary even for a hypothetical fully rigorous, scientific discipline of historical inquiry, and, in any case, narrativistic history is definitely at least partly distinct from mere story-telling (I don't know if Hayden White literally thought there was no difference at all, but he's definitely wrong if so; in fact, I would go so far as to say that there's a fact of the matter as to whether Inga Clendinnen's interpretation of the spearing of Arthur Phillip at Manly is more correct or scientific than the interpretation given by the English sources on which she relies (i.e. I don't take a super radical view on the epistemological constraints of conventional narrativistic historical inquiry, though I'm also no Geoffrey Elton)).
Now, I admit that much of what I just said, especially when I started using fancy analytic-philosophy terms, would be obscure, especially to people who don't know shit about philosophy, but in my defence, what I'm really doing is saying go read Every Thing Must Go to see why I think what I do. So go do that if you really care. Otherwise, fuck off.

A Poorly Structured, Slightly Rushed Essay on the "Indispensability Argument"

1.      Outline the Quine-Putnam indispensability argument and critically evaluate one line of attack on this argument.

The “Quine-Putnam indispensability argument” is the name given to the simplified synthesis of the similar pro-‘realist’ arguments made in the 1970s by two of the most important figures in 20^th Century philosophy – the great Harvard logician, Willard van Orman Quine, and the great Harvard philosopher-chameleon, Hilary Whitehall Putnam (1928-2016). It is perhaps the most important and widely discussed argument in the contemporary philosophy of mathematics.

In his Stanford Encyclopedia article, “Indispensability arguments in the Philosophy of Mathematics”, Mark Colyvan gives what is probably the most concise version of the argument:

Premise 1: We ought to have ontological commitment to all and only those entities indispensable to our best scientific theories.

Premise 2: Mathematical entities are indispensable to our best scientific theories.

Conclusion: We ought to have ontological commitment to mathematical entities (Colyvan, 2015 (1998), Section 1).[1]

In this essay, it is this first premise that I will be focussing on. My examination of this “Quinean Ontic Thesis” (as Colyvan calls it (1998)) will mainly centre on one particular line of attack that has been visited upon it: Penelope Maddy’s arguments from mathematical and scientific practice. I will argue that Maddy ultimately succeeds in her argument that Quine’s “confirmational holism” is incompatible with naturalism (even if my own views diverge from hers in several important ways).^[2] In mounting this case, I will be simultaneously evaluating what is probably the major response to Maddy’s argument: Mark Colyvan’s strong defence of indispensability (1998, 2001).  Ultimately, I will diverge from both Colyvan and Maddy: I will argue that Maddy’s attack does succeed in doing serious damage to the first premise, and yet I will claim that the truth-oriented (rather than ontological) versions of the indispensability argument, like Putnam’s version of the argument or Resnik’s “Pragmatic indispensability argument”, are not touched by this line of attack at all. I will thus conclude that science does give us good reason to believe that maths possess some kind of truth, but that the specifically ontological question is underdetermined by indispensability considerations.

In her 1992 paper “Indispensability and Practice”, Penelope Maddy mounts what I believe to be an ingenious case against Quine’s ontological holism.  Maddy argues that if one takes seriously the kind of commitment to “naturalism” that Quine espoused – “the claim that a philosopher can criticize scientific practice, but only on scientific grounds […] for good scientific reasons” (Maddy, 1992: 276) – Quine’s confirmational holism is itself untenable, since its overly idealised, philosophy-first rigidity ignores the way in which mathematics and science actually work.

Maddy first tries to undermine Quine’s reductionist ontology by highlighting the way in which his Ontic Thesis clashes with the behaviour of mathematicians. Although it may seem moot to draw on mathematical practice (given that Quine’s naturalism did not apply to mathematical practice), I think Maddy shows that Quine’s holism has sufficiently bizarre implications for mathematics to raise serious doubts about its reasonableness. Maddy’s basic argument in this section is that the ontological parsimony that Quine’s doctrine requires (an instance of his notorious proclivity for “desert landscapes”) forces Quine to make demarcations of mathematical ‘reality’ where mathematicians see none. Though it is, of course, necessary for any ontological realist in mathematics to make demarcations of some kind or another (one can’t say that any object imagined by a mathematician exists), I think Maddy is successful in showing the unique strangeness of Quine’s partitioning.

Maddy’s best example of the odd mathematical consequences of Quine’s doctrine comes from a statement Quine made in reply to his fellow philosopher of mathematics, Charles Parsons: “I recognize indenumerable infinites only because they are forced on me by the simplest known systematizations of more welcome matters. Magnitudes in excess of such demands, e.g. M,, or inaccessible numbers, I look upon only as mathematical recreation and without ontological rights” (Quine, 1984, 783). As Maddy points out, mathematicians working in set theory do not regard Quine’s dividing point as in any way special – in fact, they would probably think the demarcation quite bizarre.

More broadly, Maddy argues that Quine’s system simply fails to accommodate a basic fact about mathematical practice: that pretty much all mathematicians are, as Shapiro would put it (1998), “working realists” who take no notice of any considerations resembling Quine’s.[3] As Maddy herself puts it, “Mathematicians believe the theorems of number theory and analysis not to the extent that they are useful in applications but insofar as they are provable from the appropriate axioms” (Maddy, 1992: 279).

As I will argue next paragraph, I believe Maddy’s arguments are quite strong here – but I think it’s also worth bearing in mind that Quine himself would not have been convinced by them. As a point of fact, the whole basis for Quine’s confirmational holism was his strictly scientific/empirical naturalism: the belief that science is the only arbiter of ontology, and that scientific theories are the only possible foundation for a rigorous ontology.[4] It is therefore evident that Quine would not have felt obliged to expand his conception of naturalism merely to accommodate in his ontology something as wishy-washy and unempirical as the attitudes of mathematicians. Of course, Maddy wants to say that Quine’s naturalism was polluted by confirmational holism: a true naturalist, in her view, wouldn’t privilege a philosophy-first ontological doctrine over the actual practice of mathematicians and scientists (Maddy, 1992). But Maddy’s problem is that – at least with regard to Quine’s views – she is making a circular argument. Her defence of the importance of mathematical practice relies directly on claims about the “success” of mathematics as a discipline in its own right, and yet Quine saw the indispensability of mathematics in science as the singular criterion for judging mathematical success.

Since I am more open to a pluralistic naturalism, however, I ultimately side more with Maddy. Like Maddy, I believe it is correct to say that mathematics has been successful as a discipline in its own right. My reason for this is the same one that Putnam gives in his seminal paper “What is Mathematical Truth?”: the remarkable “consistency and fertility of classical mathematics”, which is explicable only if the abstract ‘territory’ which mathematicians explore has at least some objective contours (Putnam, 1972: 73). It is for this reason that I also share Maddy’s belief that we cannot ignore mathematicians’ own views on what counts as legitimate mathematics and what does not.^[5] Most importantly, I believe, like Maddy, that there is something sufficiently strange about Quine’s demarcation of mathematics to worry that there is something wrong with Quine’s ontological approach. (There is also the problem that Eliot Sober (1994) raises: mathematics is not itself empirical but a priori, and the truth of mathematical theories does not rest or fall on results in science.) Ultimately, all of this makes me think that in judging mathematics on an empirical basis, Quine is making some kind of mistake.

As I mentioned earlier, Maddy also uses examples from scientific practice to repudiate the Quine Ontic Thesis. I believe that Maddy’s scientific-practice objections to the Quinean program are even more powerful than the mathematical-practice objections, since I think these really do succeed in showing that Quine’s commitment confirmational holism leads him to violate his more basic commitment to naturalism (at least in part).

Maddy makes her case for the non-naturalism of confirmational holism through a series of examples from scientific practice – both historical and current. Her first historical recruit is the attitude of late 19^th Century scientists towards atoms. Maddy briefly describes the history: the postulated entity known as the ‘atom’ had become indispensable to the field of chemistry from about 1860 – enabling increasingly powerful predictions and calculations – and yet a large section of the late-19^th Century scientific community doubted that this entity really existed (Maddy, 1992).  Since atoms were invisible and mysterious, and since they played no role in the physics of the day (widely regarded to be more solid than the chemistry), many of the most eminent scientists of the day – including men like Ostwald and Poincaré – believed that atoms were just useful calculating devices (Colyvan, 1998: 42). The moral Maddy derives from this historical period is not hard to figure out: since confirmational holism clashes so violently with the behaviour of these scientists, confirmational holism can’t really be a ‘naturalistic’ principle.

It’s worth emphasising that Maddy’s ultimate claim here is not the simpleminded inference that we should accept any ontological view expressed by a few important scientists. Instead, she claims simply that Quine’s purist system of ontology cannot be regarded as the one measure of rationality – especially not if we claim to be scientific naturalists. In her own words, a genuine naturalist “must allow a distinction to be drawn between parts of a theory that are true and parts that are merely useful” (Maddy, 1992: 281).

As before, it is worth nothing that Quine himself would probably not have been swayed by this argument. Thankfully, we can get some sense of how Quine might have replied in the response of Quine’s mouthpiece, Mark Colyvan, in his 1998 paper “In Defence of Indispensability”. Perhaps the most noteworthy thing about Colyvan’s argument in this paper is how much ground he concedes to Maddy. Indeed, Colyvan essentially concedes Maddy’s central argument: that confirmational holism is often at odds with strict scientific naturalism. Colyvan’s main counter-claim, however, is that Maddy’s strict scientific naturalism is excessive, and does not reflect Quine’s own naturalism. This allows Colyvan to say that the contradiction Maddy supposedly identifies between confirmational holism and naturalism is really a problem she has created for herself (Colyvan, 1998: 46). On Colyvan’s view (and, he believes, Quine’s), accepting naturalism and disavowing “first philosophy” does not entail disavowing all normative philosophical systems.

When it comes to Maddy’s atomic example, Colyvan is again quite concessionary. Instead of coming to a conclusion as to what a good Quinean naturalist should say about the attitude of scientists like Poincare and Ostwald towards atoms, he merely makes three suggestions: i) that they “were making a mistake” (Colyvan’s own preference); ii) that “the controversy over atomic theory at the time gives us good reason to think that prior to 1913 chemistry/atomic theory was in a crisis period and thus the Quinean could suspend judgment on the ontological commitments of the theory”; or iii) that “given the evidence at the time it would be unwise to give total commitment to either the existence or the non-existence of atoms—some degree of belief strictly between zero and one would be appropriate” (Colyvan, 1998: 49-50). Colyvan thinks the existence of these options should be sufficient to dispel Maddy’s reservations (Colyvan, 1998: 50).

In my view, Colyvan’s response is really too concessionary here. In fact, I suspect Maddy herself would agree with the last option. This is a problem, I think, because it means that Colyvan is diluting confirmational holism to the point that it’s almost not a substantive doctrine at all (it’s almost verging on commonsense). If Quine didn’t really believe we should be ontologically committed to all and only those entities indispensable to our best scientific theories – if he instead believed that we should be ontologically committed to all and only those entities indispensable to our best scientific theories proportional to various other considerations, such as aesthetic ones – then how could he have made decisive judgments on ontological questions? Surely the point of confirmational holism is to make ontological questions clear-cut and precise – to act as a kind of ontological guillotine. If it is made vague, then it is (presumably) made useless.   

The other way in which Maddy uses scientific practice in “Indispensability and Practice” to make her case for the irreconcilability of naturalism and confirmational holism is by giving examples of applications of mathematics that are obviously not deserving of ontological commitment (on account of the obviously false assumptions they employ). Maddy’s main example of this is the average “freshman physics text”, which is “littered with applications of mathematics that are expressly understood not to be literally true: e.g., the analysis of water waves by assuming the water to be infinitely deep or the treatment of matter as continuous in fluid dynamics or the representation of energy as a continuously varying quantity” (Maddy, 1992: 281). The indispensability of such unrealistic mathematical tools is not in dispute, and yet only an insane person would think that that fact is sufficient to ontologically commit us to the mathematical objects required by these mathematical tools.

I think Maddy’s attack here succeeds in inflicting yet more damage on Quine’s confirmational holism, though I also believe that Maddy misses an opportunity in this passage to invoke the plentiful applications of unrealistic mathematics in economics – a field which did (surprisingly) fall under Quine’s very broad conception of science (Quine, 1995: 49). As anyone who has read a freshman economics text realises, neoclassical economics uses highly simplified, toy models of the real world, and typically eschews dynamic modelling – instead making totally unrealistic assumptions about tendencies towards “equilibrium”. And just as with physics, nobody would say that we should be ontologically committed to the mathematical entities needed for these models.

The way Colyvan replies to this latter class of Maddy examples in “In Defence of Indispensability” is, again, not to deny their validity, but simply to claim that scientists often do worry about the ontology of the mathematics they are using – thus proving that it is certainly not anything goes. Colyvan’s example of useful maths that was viewed with suspicion by scientists is the Dirac delta function. This strange mathematical object proved very useful in quantum mechanics, and yet, as Colyvan notes, it “attracted much criticism” (Colyvan, 1998: 52).

It’s worth nothing that Colyvan is once again making massive concessions here. He has strayed so far from the literal Quine Ontic Thesis and the literal doctrine of confirmational holism that he is looking for evidence of scientists not taking confirmational holism too seriously (by being suspicious of the ontology of indispensable mathematics) in order to show that some enervated version of confirmational holism is still possible. It’s a strange position, in my opinion.

All in all, my conclusion about Maddy’s attack on the first premise of the Quinean indispensability argument is that it is broadly successful. As I’ve made clear, I think Maddy’s success is even evident in Mark Colyvan’s attempt to defend the premise, since Colyvan’s defence involves diluting the doctrine of confirmational holism to such an extent that it basically loses all meaning. Though I haven’t mentioned this philosopher so far, I also think that Maddy’s critique is boosted by some of the observations of Stephen Yablo (1998) about the ineliminably metaphorical nature of a lot of the language in our best scientific theories. As Yablo notes, Quine never adequately acknowledged the importance of figurative language in our scientific discourse, and this is reflected in the rigidity of the doctrine of confirmational holism.[6]

As I suggested in my introduction, I certainly don’t think that there isn’t some salvageable form of the indispensability argument. Indeed, I think the truth-oriented versions of the argument are totally unassailable. One example of such an argument is Michael Resnik’s “Pragmatic Indispensability Argument”, laid out in his book Mathematics as a Science of Patterns. Resnik’s formulation doesn’t rely on confirmational holism, and thus doesn’t have a premise nearly as strong as the Quine Ontic Thesis. Instead, its two key premises are that “We are justified in drawing conclusions from and within science only if we are justified in taking the mathematics used in science to be true” and “We are justified in using science to explain and predict” (Resnik, 1997: 47). The idea is that science’s success only makes sense if the mathematics used in it is largely or wholly true.

I personally see no way of denying this argument. As I made clear in footnote 1, Putnam also made a similar truth-oriented argument, and was generally less rigid about ontology than Quine, holding no such doctrine as confirmational holism. This makes his indispensability arguments  more attractive, in my view.

Bibliography

Colyvan, Mark (1998). “In Defence of Indispensability”, Philosophia Mathematica, 6 (1): 39-62.

(2001). The Indispensability of Mathematics, Oxford University Press.

Maddy, Penelope (1992). “Indispensability and practice”, Journal of Philosophy, 89 (6): 275-289.

Putnam, Hilary (1975). “What is Mathematical Truth?”, in Mathematics, Matter and Method, Cambridge University Press 60--78.

Quine, Willard van Orman (1961). “On What There Is”, in Tim Crane & Katalin Farkas (eds.), From a Logical Point of View, Harvard University Press, 21-38.

(1995). From Stimulus to Science, Harvard University Press, Cambridge, Mass.

Resnik, Michael (1997). Mathematics as a Science of Patterns, Oxford University Press.

Shapiro, Stewart (1996). The Philosophy of Mathematics: Structure and Ontology, Oxford University Press.

Yablo, Stephen (1998). “Does Ontology Rest on a Mistake?”, Aristotelian Society Supplementary Volume, 72 (1): 229 - 283.

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[1] It should be noted that this formulation is very Quine-biased, and doesn’t very well capture Putnam’s indispensability arguments. In fact, Putnam criticises Colyvan’s formulation directly in chapter 9 of the 2012 book Philosophy in an Age of Science: “From my point of view, Colyvan’s description of my argument(s) is far from right” (Putnam, 2014: 1).  Putnam’s objections to the formulation are several. Firstly, it doesn’t capture the important distinctions between his and Quine’s views: “The fact is that in “What is Mathematical Truth” I argued that the internal success and coherence of mathematics is evidence that it is true under some interpretation, and that its indispensability for physics is evidence that it is true under a realist interpretation” […] “a distinction that Quine nowhere draws” (Putnam, 2014: 2).  Secondly, Putnam points out that he never argued for anything as straightforward as “ontological commitment” to objects or entities (and was never a “Platonist” per se): in both “What Is Mathematical Truth?” and “Mathematics Without Foundations”, he “said that set theory did not have to be interpreted Platonistically” and that “modal-logical mathematics (i.e. mathematics which takes mathematical possibility as primitive and not abstract entities of any kind) and mathematics which takes sets as primitive are “equivalent descriptions”” (Putnam, 2014: 2). Whereas Putnam’s “indispensability” argument was – as he puts it – “an argument for the objectivity of mathematics in a realist sense—i.e. for the idea that mathematical truth must not be identified with provability”, Quine’s indispensability argument was an “argument for “reluctant Platonism,” which he himself characterized as accepting the existence of “intangible objects” (numbers and sets)” (Putnam, 2014: 2). Finally, Putnam never subscribed to the “and only” part of the first premise (Putnam, 2014: 3).

Despite all these problems with the formulation, it is Colyvan’s exposition of the argument that I will be attacking in my essay. One good reason for sticking with this Quine-biased formulation is that most of the literature on the ‘Quine-Putnam indispensability argument’ is also Quine-biased.

[2] ‘Confirmational holism’ is a slightly misleading name for Quine’s mechanical system of ontology. In essence, it refers to Quine’s claim, famously expressed in his essay “On What There Is”, that a scientific theory  is “committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true” (Quine, 1943: 33).

The key feature of this doctrine is its rigidity. According to Quine, if a new grand theory of physics were to come along that bettered our current Relativity/quantum mechanics synthesis in terms of explanatory accuracy, predictive power, ontological parsimony (and so on), we should immediately abandon all ontological commitments to entities that aren’t necessary for the truth of the affirmations of the new theory and immediately adopt ontological commitments to all and only the entities that are. (It’s clear why I called it a “mechanical system of ontology”.)

[3] Hilbert was unusual (and his program failed anyway).

[4] We will later see that Colyvan, a neo-Quinean, also defends this view (1998, 2001).

[5] Note, however, that I am not endorsing Maddy’s own ontology. I have as yet said nothing about whether any mathematical “objects” “exist”. 

[6] As Yablo writes in “Does Ontology Rest on a Mistake?”, “Intentional attributions, subjunctive conditionals, and so on are said [by Quine] to have ‘no place in an austere canonical notation for science’ suitable for ‘limning the true and ultimate structure of reality’”.