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There was a lot in the Search on Russell, so I will continue my notes mainly on Russell. Read the rest of this entry »

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A large part of the Search for Mathematical Roots focuses on Russell and the development of Principia (PM, hereafter). I found these chapters, which roughly comprise the latter half of the book, to be quite helpful since I’m less familiar with Russell than Frege and Wittgenstein and the chapters do a good job of explaining the influence of PM. In an interesting bit of trivia, Grattan-Guinness says that its name is a nod, not to Newton’s book, but to Moore’s Principia Ethica, Moore being a huge influence on Russell’s philosophical development. Grattan-Guinness makes Russell out to be heavily influenced by Peano. In the historical narrative, Russell’s interests seem to change along the same lines that Peano’s do. An exception to this is that Russell maintains that math is a part of logic whereas Peano thinks they merely overlap. Peano even wrote a paper on “the” in which he gave the same principles for its meaning that Russell did in “On Denoting.” According to Grattan-Guinness, Russell seems to have been familiar with that paper, at least reading it once, but seems to have forgotten about it.

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Here are some more thoughts on Wittgenstein on the foundations of math. For those interested, be sure to check out the prose interpretation of Wittgenstein on math and games at Logic Matters. In part 6 of the Remarks on the Foundations of Math, Wittgenstein presents several thought experiments to probe a cluster of notions, calculation, proof and rule. One of these is two-minute England in section 34.

Two-minute England is described in the following way. God creates a country in the middle of the wilderness that is physically just like England. The caveat is that it only exists for two minutes. Everything in this new country looks like stuff in England. In particular, one sees some people doing stuff that exactly mimics what English mathematicians do when they do math. This person is the one to focus on. Wittgenstein asks, “Ought we to say that this two-minute-man is calculating? Could we for example not imagine a past and a continuation of these two minutes, which would make us call the process something quite different?”

The questions, I take it, indicate doubt that we must take the two-minute-man as calculating. We can, but it is not compulsory. This is because there is no reason, given what we’ve observed, to think that he must be calculating. In this case there is no fact about it. We might be tempted to attribute calculation to the two-minute-man because we fill out his story with some events leading up to and following this that lead us to think that he is calculating. These events don’t happen, since he only exists for two minutes. There is no wider context for this person that settles whether they were calculating, or scribbling, or regurgitating symbols seen elsewhere.

The point of this thought experiment is to present some evidence that calculation is not identifiable with any bit of mere behavior. Connecting this with other sections of part 6, the behavior only becomes calculation when it is connected up with some appropriate purposes or situated in a normative context in which it is appropriate to talk about correct or incorrect calculation. Wittgenstein’s focus is to argue that various mathematical notions are like this.

This passage is preceded by one in which Wittgenstein says, “In order to describe the phenomenon of language, one must describe a practice, not something that happens once, no matter of what kind.” The two-minute England thought experiment is intended to illustrate this point. I’m not sure that there is anything in the two-minute-man’s life that would let us embed it in a practice of some sort.

Connecting the thought experiment up in a nice way with what precedes it requires fleshing out the notion of a practice, which I can’t do. There are scattered remarks on that idea in the Remarks, which I haven’t begun to put together. Despite this, it seems to me that two-minute England fits together better with what comes earlier and later in this part of the Remarks, namely that calculation isn’t just a matter of behavior. This needs to be connected with the wider concern of what it means to follow a rule in order to make it a bit clearer, I think. Incidentally, I think that the rule-following discussion in the Remarks is more accessible than the discussion in the Philosophical Investigations.

I’m reading through some of Wittgenstein’s lectures on the foundations of mathematics. I’m not real sure what to expect. I thought I’d write up some notes on it as I went along. This is, in my opinion, the only way to read Wittgenstein. I figured I’d post them in case anyone can help shed some light on what is happening or is interested. Read the rest of this entry »

In MacFarlane’s thesis, he distinguishes three related but distinct notions of formality that have been important in the evolution of the conception of logic, 1-formality, 2-formality, and 3-formality. The first is defined as being normative for thought as such. The second is defined as being insensitive to distinctions amongst objects, usually cashed out in terms of permutation invariance. The third is defined as abstracting from all conceptual or material content. In Kant these three notions are equivalent due to his other commitments, notably theses connected to his transcendental idealism. In Frege, the first and third come apart and Frege thinks the second does not characterize logic. Tarski and those writing after him focus mainly on the latter two and it seems that the second has been given pride of place since it admits of such a crisp mathematical formulation. Read the rest of this entry »

I was thinking about the Tractatus recently and came up with a question about it that I was unsure about the answer to. (This is not that hard to do really.) The question is whether Wittgenstein thinks that it is a logical impossibility that there could be no objects.

Why would one think that this is not an option that there could be no objects? In the 2’s, Wittgenstein talks about how there must be a substance to the world, and this substance is comprised of Tractarian objects. This makes it seem like it is not a logical possibility for there to be no objects. Granted, this would only be a logical possibility if the world were connected tightly to language or logic or if objects were similarly tightly connected to language. I’m not terribly comfortable with the 2’s, either alone or together with what comes later. Luckily, I don’t think that we need to appeal to them specifically in order to come up with an answer, which point I’ll get to below.

Why would one think that it is an option? In 5.453 Wittgenstein says: “All numbers in logic must be capable of justification. Or rather, it must become plain that there are no numbers in logic. There are no pre-eminent numbers.”
If it is a matter of logic that it is impossible for there to be no objects, this would seem to make zero a distinguished or pre-eminent number, which seems to be ruled out by the above. It might look like I’m running together objects and names, and all that logic will deal with is the names. In the Tractatus, however, every name designates an object.

Alternatively, one might ask why there must be at least one thing. Is this asking for logic to give a justification? Thinking about the way that the TLP is set up, it seems not. Rather, this issue is left implicit in the propositions. Propositions consist of concatenations of names. Names designate Tractarian objects. Thus, for there to be Tractarian propositions at all, there must be names and so objects. In order to be talking about logic at all, we must presuppose that there are at least some objects. It looks like the question of why there is something rather than nothing is barred from the outset, which is probably something that Wittgenstein would’ve approved of.

I do want to note that things are complicated with Tractarian claims about possibility. In TLP, the objects are the same in all possible worlds. Indeed, the possible worlds, if we want to use that language, are constituted by those things in various arrangements of facts. It seems like claims such as “there could have been more or fewer things than there are,” if formulable in a Tractarian proposition, must come out false. The possibilities are completely determined by the Tractarian objects that there actually are. Of course, the way that the above claim is formulated, in terms of generic things, is probably the source of this seeming weirdness. “Thing” and “object” are formal concepts in the TLP. A claim like “there could have been more espresso cups than there actually are” needn’t turn out necessarily false because “espresso cup” is a proper concept, which can be expressed with a propositional function.

In Logical Syntax, Carnap says that he has shown that one can talk about the logical form of language and that this is a counterexample to Wittgenstein’s dictum that you cannot talk about logical form, as it can only be shown. There is something that seems odd about this. From the little bit of secondary literature I’ve read, no one really seems to say much about this, although some of Carnap’s contemporaries seem to embrace Carnap’s claims. It seems like Carnap is talking past Wittgenstein. The problem with fleshing out this claim is that I have to flesh out one of the difficult doctrines in the Tractatus. (As opposed to the simple ones, I guess.) I’m going to attempt to sketch an answer. Read the rest of this entry »

A couple of weeks ago Warren Goldfarb gave a short talk to our TLP reading group on the notion of showing. Explaining the saying/showing distinction is very important to resolute readers of the Tractatus and it is one of the big ways in which the resolute reading is set apart from the others. The traditional reading takes the talk about showing to indicate that there is some inexpressible reality or truth that statements can gesture towards but not outright say. While these things are nonsense, they are informative nonsense in that they help us see the deep truths. The resolute reading wants to do away with that and say that nonsense is just nonsense. Alright, but what do we make of various places in TLP where Wittgenstein says things like what can be shown cannot be said?
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I’m participating in a TLP reading group this term. We’re going to make it through the end of the 3’s by winter break. One point that I’ve made several times to individuals, and which came up again in discussion on Friday, is something that I thought would be worth putting online. It is originally due to Michael Kremer, in his excellent “Mathematics and Meaning in the Tractatus.” Many people think that in TLP, there is a very strict two part distinction: sentences have sense while names have meaning (Bedeutung). Names have no sense; they just mean the objects to which they refer. The objects are the meaning of the names. This is a pretty natural reading to get out of the 3’s. However, sentences also have meaning. Wittgenstein uses “meaning” in two ways throughout TLP. Sometimes he means the stricter sense to characterize the relation between names and objects, and sometimes he uses it in a more general way to talk about whatever significance linguistic units have. The textual evidence for this comes from 4.4241: “When I use two signs with one and the same meaning, I express this by putting the sign ‘=’ between them. So ‘a = b’ means that the sign ‘b’ can be substituted for the sign ‘a’.” The signs used there are the ones normally used as names, lower case letters from the early part of the alphabet. However, combine this with 5.254: “An operation can vanish (e.g. negation in ‘∼∼p’ : ∼∼p = p).” We clearly have propositional signs flanking the ‘=’. This means the two signs have the same meaning. However, they aren’t names. Therefore, Wittgenstein has two senses of meaning in place in TLP. This is a fairly straightforward point but it is missed by a lot of people.

I didn’t realize how odd the preface of the Tractatus is until last week during the PItt reading group. In particular, the following:
“If this work has a value it consists in two things. First that in it thoughts are expressed, and this value will be the greater the better the thoughts are expressed. … On the other hand the truth of the thoughts communicated here seems to me unassailable and definitive.”
I think the latter bit, after the ellipsis, gets the most focus usually. The truth of the thoughts is important. But, what struck me in a way it hadn’t before was the first bit. The work has value insofar as it expresses thoughts? That seems to set the bar low as it isn’t that hard to express a thought. The interesting thing is combining this with 6.54, which says that anyone who has understood the propositions of the book will recognize them as nonsense. This would make it a little harder to take Wittgenstein as expressing thoughts. In the preface he says that he doubts he has done it well. This might be literary self-deprecation, but that seems a bit unlike Wittgenstein.

We should note that Wittgenstein doesn’t say what thoughts are, or are supposed to be, expressed in the book. Just that some thoughts are. The truth of these thoughts he thinks is clear. Michael Kremer says some interesting things about different senses of truth in his paper on solipsism that I suspect are important for understanding Wittgenstein here. In short, he thinks there is a non-propositional sense of truth. This is not the ineffable sort of truths that some realists ascribe to the TLP. It is more like the sense of truth expressed when people say things like “the truth in beauty” or “the truth in solipsism” (to use Kremer’s title). Taking the preface to be meaning truth in this sense would go some ways towards making it consistent with the end of the book. The problem would probably come from the expression of thoughts. This would, it seems, have to be thoughts in a sense distinct from the Tractarian view of them, i.e. as significant propositions. Otherwise, taking a non-propositional view of truth would be a non-starter. There isn’t a corresponding idea of thought developed in Kremer’s paper. He says some possibly relevant things about solipsism among other “ways of thinking” which might be fleshed out appropriately, but it would result in interpreting the preface in such a way that it resembles the body of the book very little. That might not be a bad thing though. I don’t have a well-developed idea here, but I think there is some promise to making sense of the preface along these lines.

(Running a quick search on Kremer’s article, it seems that he talks about the preface. However, he doesn’t talk about the part I am talking about. He concentrates on the early part of the preface which discusses drawing a limit to thought.)

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

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