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In his article “Present State of Research into the Foundations of Mathematics,” Gentzen briefly talks about Goedel’s incompleteness results. He says that it is not an alarming result because it says “that for number theory no adequate system of forms of inference can be specified once and for all, but that, on the contrary, new theorems can always be found whose proof requires new forms of inference.” This is interesting because Gentzen worked with Hilbert on his proof-theoretic projects and created two of the three main proof-theoretic frameworks, natural deduction and the sequent calculus. The incompleteness theorems are often taken as stating a sort of limit on proof-theoretic means. (I’m treading on shaky ground here, so correct me if my story goes astray.) That is to say, any sufficiently strong proof system will be unable to prove certain consequences of its axioms and rules. Adding more rules in an attempt to fix it can result in being able to prove some of the old unprovable statements, but new ones (maybe just more?) statements will arise. Read the rest of this entry »

Before getting to the post proper, it will help to lay out a distinction drawn, I believe, by Sellars. The distinction is between three sorts of transitions one could make in relation to propositions, for example if one is playing a language game of some sort. They are language-entry moves, language-language moves, and language-exit moves. The first is made through perception and conceptualization. Perceiving the crumb cake entitles me to say that there is crumb cake there. The second is paradigmatic inferential or consequential relations among propositions. Inferring from p&q to p is a language-language move. The third is moving from a practical commitment or explicit desire to action. Borrowing Perry’s example, it is the move from thinking that I have to be at the meeting and that the meeting is starting now to me getting up and rushing off to the meeting.

In Making It Explicit, Brandom distinguishes three things that could be meant by inferentialism. These are the necessity of inferential relations, the sufficiency of inferential relations, and hyperinferentialism. The first is the claim that inferential articulation is necessary for meaning. Representation might also be necessary, but at the least inference is necessary. The second is the claim that inferential articulation is sufficient for meaning. In both of these, inference is taken broadly so as not to collapse into hyperinferentialism, which is the thesis that inference narrowly construed is sufficient for meaning. The narrow construal is that inferences are language-language moves. What does this make the broad construal? According to Brandom, it includes the language-entry and -exit moves. In MIE, Brandom defends, I believe, the necessity of inferential relations, although he says some things that sound like he likes the idea of the sufficiency claim. He doesn’t think that hyperinferentialism will work. This is because he thinks that for some words, the content of the word depends on causal/perceptual connections. I think that color terms are examples. Additionally, the content of some words exhibits itself in what practical consequences it has in our action and this exhibition is an essential part of the meaning of the word. My beliefs about crumb cake will influence how I act around crumb cake. Hyperinferentialism cannot account for these because the language-entry and -exit moves essential to their meaning are not things hyperinferentialism has access to.

Brandom’s claim then, once things have been unpacked a bit, amounts to saying that the narrowly inferential connections, perceptual input, and practical output are necessary for meaning. This seems to undercut the charge that inferentialism loses the world in a froth of words, which charge is mentioned at the end of ch. 4 of MIE, I think. It is also a somewhat looser version of inferentialism since things that are not traditionally inferential get counted as inferential. The inferentialist could probably make a case that that the language-language moves are particularly important to meaning, but I think Brandom’s inferentialism stretches the bounds of inference a bit. I’m not sure an inferentialist of the Prawitz-Dummett sort would be entirely comfortable with the Brandomian version of it. By the end of MIE, Brandom’s broad notion of inference encompasses a lot. Granted, it is fairly plausible that much of that is important to or essential for meaning. However, I wonder if it doesn’t move a bit away from the motivating idea of inferentialism, namely that inference is what is central.

A formulation of mathematical platonism is that mathematical entities exist independently of us and when we do math we are exploring the realm of the mathematical entities and discovering new things about them. This is a fairly basic or naive formulation of it, but it gets the flavor roughly right. A software platonist would think that programs exist independently of us and when we do computer science or programming (not sure which would be the better formulation) we are exploring the realm of programs and discovering new things about them. A mathematical platonist would say that the natural numbers would exist even if humans never did and a software platonist would say, e.g., that LISP or Apple OS X would exist even if humans never did. Further, the natural numbers were around long before there were humans, and, similarly, LISP and Apple OS X were around long before there were humans. We started exploring the natural numbers a long time ago but only recently started exploring LISP and even more recently Apple OS X.

It seems to me that software platonism is nuts. There is a fairly strong sense, I would think, in which Apple OS X would not have existed were we (broad, inclusive ‘we’) not around to create it. It was created by engineers and is changing over time. But, it is still just a program. However, as such, there is a Turing machine equivalent to it. Now, LISP was also created by a computer scientist, but it is a bit more abstract and so the platonist intuitions are maybe stronger for it. Engineers have changed LISP over time, and I don’t know how a software platonist would explain what links the various versions of LISP (or, for that matter, how programs of any sort run on hardware, but that might not be so different than us counting with the natural numbers). LISP has, I believe, been shown to be able to compute the same functions as Turing machines; the term is Turing-complete, I think.

The question is: are we creating or exploring an antecedently existing realm of programs? There is a related question: are we creating or exploring an antecedently existing realm of Turing machines? I’d hope the two fall together, given the tight connection between programs and Turing machines. However, there seems to be a little bit of room to drive a wedge between the two. I suppose that would allow one to be a Turing machine platonist without being a software platonist. Although, one could try to use the Turing machines, which may be able to draw on mathematical platonist leanings, to argue for platonism about programs.

I don’t have any worked out ideas in this area, but it does seem to me that software platonism is hopeless. Are there any software platonists?

Thinking about some things in the Tractatus together with things I’ve been reading for other classes (mostly the Moral Problem) has led me to think about clarity in writing again. It is still somewhat elusive. One of the virtues of analytic philosophy, I have been told, is that the writing is clear and the arguments are explicit. The importance of clarity in writing seems to me more and more odd. There are many things one might be looking for in reading a philosophy book or article. One of these is good philosophy. There isn’t much reason to see clear writing as particularly indicative of good philosophy (an amorphous term which won’t be defined here). Placing a high premium on clear writing could lead to the following two sorts of errors. One is the error of mistaking clear writing for good philosophy. The other error is mistaking unclear writing for bad philosophy.

One could take the Searle approach and claim that unclear writing is indicative of poor understanding. There is something right in this. For a lot of cases, this is true. It seems like a good heuristic for grading undergraduate papers. As an across the board maxim, it seems likely to lead to error. There are a lot of people who understand abstract subjects very well but are bad at communicating this to others. It isn’t particularly hard to think of TAs who clearly understood things but could not explain them appropriately. Similarly, a presentation might be unclear because the presenter is loading the presentation with so many qualifications and subtleties as to lose the audience. But, this is just because the presenter is being very careful, overly so. At a certain level, it doesn’t make sense to hold to Searle’s maxim. Clarity of presentation becomes something of an accidental feature rather than indicative of understanding. As another example, some people, I hear, accuse John McDowell of being somewhat opaque, but I would be extremely hesitant to make the leap to saying that this is because he doesn’t understand what he is talking about. Doing that seems like a reductio of the Searle maxim.

Going the other way, clear writing isn’t always indicative of good philosophy. It is certainly less frustrating to read an article that is well structured so that the dialectic is easy to follow than it is to read an article that is not well structured. The danger is that the author gets caught up making the article clear at the expense of focusing on the philosophical issue, e.g. explicitly naming all the principles, numbering all the steps of the argument, using variables in different scripts for terms whose particular values don’t matter, etc. These things all help keep one honest, and I am a fan of them. I use them a fair amount in writing my own papers. But, they go a ways toward giving an illusion of rigor that can lead the reader/author into a false sense of security. If clarity by itself meant philosophical issues were on full display and under control, then philosophy of logic and philosophy of math would not exist, it seems. (That is probably a bit of a hasty assertion but I’m going to leave it for now.)

To head off a worry (possibly a worry only I have about what I wrote), I don’t want to endorse any kind of deep vs. (merely) clever dichotomy, something which Wittgenstein himself apparently endorsed (a fact which saddens me). I’m not sure these have to be seen as a dichotomy or even in opposition. Philosophical merit is one thing, and stylistic merit is another. Sometimes the two go together, as in David Lewis. Sometimes they come apart, as in… a lot of philosophers. Confusing one for the other seems like a bad move. Trying to use one to cover for the other seems like an even worse move. There are myriad benefits to writing clearly, and there are reasons enough to keep teaching and endorsing that practice.

The way forward in thinking about this is, probably, to hack away at the idea of clear writing for a bit. I’m not going to attempt to go at the idea of good philosophy. I mean, really, that would be pure hubris.

In Wittgenstein, Tom Ricketts said something that reminded me that I need to write up a few more posts on Through the Looking-Glass. I’ll quote the important bit:
“‘You needn’t say “exactly,”‘ the Queen remarked. ‘I can believe it without that. Now I’ll give you something to believe. I’m just one hundred and one, five months and a day.’
‘I can’t believe that!’ said Alice.
‘Can’t you?’ the Queen said in a pitying tone. ‘Try again: draw a long breath, and shut your eyes.’
Alice laughed. ‘There’s no use trying,’ she said, ‘one can’t believe impossible things.’
‘I daresay you haven’t had much practice,’ said the Queen. ‘When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast. There goes the shawl again!'”

If we grant that believability is sufficient for conceivability (which strikes me as prima facie plausible), then Carroll endorses the idea that conceivability does not entail possibility. However, we can still conceive of and believe in impossible worlds (although Carroll doesn’t go in for worlds talk). Don’t think we can? The White Queen replies: just try harder; maybe shut your eyes and breath deeply. It seems to me, (and this may get overly geeky), that Carroll would further reply to someone who says that such and such a scenario is inconceivable, in much the same way as Inigo Montoya: “You keep using that word. I do not think it means what you think it means.”

Since I’m taking a class on the early Wittgenstein, I’ll probably have a few posts this semester on issues from the Tractatus. I figure I’d kick things off with a few thoughts on an issue that I didn’t think I’d find in the Tractatus. Wittgenstein had some thoughts on clarity. Amazingly, the Routledge edition of the Ogden translation has an index entry for clarity. Proposition 4.1 says “A proposition presents the existence and non-existence of atomic facts.” From this, he goes on to comment at 4.112 “The object of philosophy is the logical clarification of thoughts. Philosophy is not a theory but an activity. A philosophical work consists essentially of elucidations. The result of philosophy is not a number of ‘philosophical propositions’, but to make propositions clear. Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred.” This is getting on the train of thought that I’m interested in. It is made explicit in 4.116 “Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.”

The important difference between this Wittgenstein and Rorty probably comes down to what is meant by ‘clearly’. When Wittgenstein says anything that can be said or thought can be done so clearly, who does he think can say or think these things clearly? Is it the person who initially says or thinks them? That is somewhat dubious since there are certainly things that one could think without quite getting clear on what exactly they are getting on. But, maybe he means that someone, some sympathetic interpreter, can put the thoughts clearly. To hop between topics somewhat, this seems to be part of the project of Brandom’s Making It Explicit, Articulating Reasons, and Tales of the Mighty Dead. It is part of the project that interpreters make explicit collateral beliefs and commitments of whomever is being interpreted so that the trains of inference can be fairly assessed and criticized. The basic premise of 4.116 seems to be in the background of Brandom’s approach. Whatever can be thought can be made explicit, and, through the supplementation of collateral commitments made clear. But that is sort of an aside. I find 4.116 fairly plausible, although that belief comes and goes depending on what I’ve been working on.

The remarks in 4.112 are also kind of appealing. Although I’m not sure whether we should take the elucidations and clarity to be analysis in the sense of conceptual analysis. Now, in what sense does philosophy make other propositions clearer? is the job to make clear the dialectical structure used? This would restrict the application of philosophy to things that have an argumentative structure. Maybe the job is to try to draw out the conceptual relations between ideas that are used in various disciplines or texts. That sounds somewhat like traditional philosophical projects. Wittgenstein seems to want to draw a distinction between elucidations and philosophical propositions, but if the job of philosophy is just to produce elucidations, then it would seem like elucidations are philosophical propositions. It seems reasonable to take him to mean that philosophy does not result in traditional philosophical propositions, rather explanations. This is somewhat attractive, although I think I disagree with it. There seems to be some substantive philosophical propositions out there, e.g. Dummett’s anti-realism, which are not merely explanatory, but I imagine that Tractatus Wittgenstein would say those are nonsense.

Continuing a series of practically content-free posts, here’s a great quote from Dummett’s preface to his Logical Basis of Metaphysics, which finally showed up at Pitt’s library.

“We all stand, or should stand, in the shadow of Wittgenstein, in the same way that much earlier generations once stood in the shadow of Kant; and one of my complaints about many contemporary American philosophers is that they appear never to have read Wittgenstein.”

[Update: Follow ups here and here.]

Richard Rorty once criticized analytic philosophers for dismissing writing that was unclear as being a product of confusion or subterfuge. He went on to say that he had never seen an argument for the conclusion that clarity entails or promotes philosophical depth or insight. After giving it some thought, I don’t think I’ve seen an argument for that either. There are a couple of reasons for this. It is a metaphilosophical point about which not that many people write, as far as I can tell. It is taken as a presupposition by a lot of people that clarity aids or entails insight. Why should this be though? If I have written a clear paper with a valid argument that is easy to follow, then it might be easier to read. This is not enough for insight, depth, or progress. The content of the argument needs to be interesting and lead to interesting places. Is there any reason to think that an argument laid out in a straightforward deductive style with all the premises explicit will facilitate interest? It is plausible that an argument laid out with some premises implicit, not necessarily with the conclusion at the end, could facilitate interest. If the argument reflects the thought processes of the author, it is quite possible that a lot of the argument had to be constructed after the fact to bridge dialectical gaps that emerged in the course of writing. This could all be done with a fair amount of clarity though. Why think that someone who is less than crystal clear, e.g. Heidegger, is not as deep as someone who is fairly clear, e.g. Quine (at times)? Suppose we take as our model of philosophical argumentation first-order logic, or some rigorous proof theoretic system. These are models of clarity; there is nothing hidden in their notation. If these are our models, then anything that requires close textual interpretation is already deviating from them. Those deviant texts have hidden premises and implicit assumptions. Switching gears some, we could take as our model scientific papers. I’m less familiar with scientific journals than with math and logic journals. Assuming they are fairly similar to computer science journals, then they will be fairly clear. The occassional metaphor might be used to illustrate a point, but not much else. There won’t be any exigetical heavy lifting required. Deep results in science can be obtained without appealing to unclarity. However, some science writing is bad and is a bit hard to follow. This might undermine the point a little, but such papers still don’t require the expertise of textual interpretation that Kant or Hegel do.

Here’s another stab at it. If someone has written something that is difficult to follow and somewhat unclear, then we think that if they understand it well and there is some substance to it, they will be able to reformulate it in a form that we can better follow. If this is our implicit principle, then it says something about us too. The author should be able to reformulate her idea in such a way that the reader can more easily follow her thoughts. That’s what it says about us. Following this principle as an author makes things easier on the reader, but again, it doesn’t seem to have anything to do with depth of thought. Maybe it belies a bias towards extensionalism. If arguments/books/papers have a content which is extensional, then that content can be specified in different ways that are all extensionally equivalent. In the case of unclear writing, the author has selected one way whose presentation makes things difficult for us. She should be able to reformulate it in an extensionally equivalent, more accessible form. If this bias is dropped, arguments/books/papers could be said to intensional in that their content depends on its presentation in some important way. This is likely to be the case when an idea is put in an unusual way in order to draw attention to it and make the reader abandon some particular connection with that idea or presupposition. Possibly, once the reader has drawn out the moral she may kick away the intensional ladder and keep the extensional content, but that would only be once she’s figured it out. The form of the presentation was crucial to reaching the insight. Some writers, like Wittgenstein, might use unusual forms or presentations to drive home this point.

I guess my tentative conclusions at this point are: (i) that the presupposition that clarity facilitates depth/insight is probably derived from taking science/math/logic as a model of inquiry, (ii) clarity is more an aid to the reader than the writer, and (iii) the presupposition might be false in some cases, e.g. Wittgenstein.

Quine said that philosohy was on a continuum with science. He also said, “Philosophy of science is philosophy enough.” I’m not sure if I’ve heard a good explanation of the way(s) in which philosophy is different only in degree and not in kind from scientific practice in general. Here are a couple of stabs at what that could mean. First, he could have meant that there is no first philosophy above science and all philosophy should come from science. This might not be too far off. Another way of taking it is that philosophy should be commentary on scientific results and the practice of scientists. This would explain the philosophy of science quip. A third way of taking it is that science is the regimenting of common sense practices of testing and forming beliefs to make sense of the evidence and philosophy is also a method of systematizing common sense and bringing it into reflective equilibrium. This would be a kind of David Lewis idea of philosophy. I don’t think this is exactly what Quine had in mind, but it meshes well with his idea.

There are at least two theories as to the main evolutionary benefit of language. (This is not the best way of putting it, but I think it captures how I remember each side.) One side thinks that the big advantage of language is communication. With language you can discuss things with others and exchange contentful ideas. The other side thinks that language has nothing to do essentially with communication and it is rather mental organization that is the main benefit. Language use adds a large amount of structure to the mind and it is what enables us to have the abstract thoughts that we do and make long-term plans involving various contingent and hypothetical possibilities. The communication stuff is a happy accident of this. The latter position is defended by Chomsky and his followers. The prior position is defended by pretty much everyone else I think. There is a prima facie harmony between the views (as indicated by my comment following the second description). One view says language is for exchanging contentful messages and the other view says it is for having contentful thought. This meshes with the idea that language is used for putting into a public medium the content of one’s private thoughts. (That probably sounds quite naive and would not make partisans of either side happy.) I started thinking about whether there are any other possible pressures for the development of language or whether these two options exhaust the range of viable options. My initial instinct is that there should be more possibilities although I haven’t come up with them.

I had more thoughts about how to approach language use that may seem naive. One thread in the philosophy of language sees language, particularly semantics, as a logical phenomenon. Logic is the primary tool for investigating and understanding language. As Richard Montague put it, syntax, semantics and pragmatics (!!) are all mathematical phenomena. Another way of thinking about it is as a completely naturalistic phenomenon that arose in a species that developed under evolutionary pressure. Pressures that resulted in language development should have resulted in various processing limitations as well in order to be adaptationally useful. Continuing this idea: tools that are useful for understanding evolutionary development should be useful for understanding restrictions on languages, in particular, (getting into uncertain territory here) information theory, noisy-channel models, and models of redundancy. Thinking about it, it is kind of amazing that evolution ended up giving us logic, to put it one way.

One other thing that I feel gets left out of the logical models of language is the fact that while languages might have a logical basis of some kind, they also have a historical basis. They are entities (for the moment begging the question about the existence of public languages) that have developed in haphazard ways in response to various contingent, historical developments that could have easily gone other ways. There are words and phrases whose meanings have shifted over time and parts of languages whose syntax has changed dramatically. These elements are all thrown together, at least in languages that have live modern versions. These considerations make the idea of a single, synchronically pure language an idealization. That is to say, I’m not particularly surprised that we don’t have grammars for any natural language that generate all and only the sentences of that language.


Shawn Standefer, recent Ph.D. in philosophy from Pitt. (More about me)