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I have been reading Wittgenstein's Philosophical Investigations and my question is how does he come to realize that we can't have a perfect language.

For instance I would say math is a perfect language because one can never mistaken 5 to mean 3 or addition to mean subtraction. Math in my opinion, perfectly represents the universals as well as the particulars, which is why I think there is never any misrepresentation.

So in his words why would he say we can't have a perfect language?

Note: I think its because he thinks language is used as a tool and is never exact, that it is all dependent on the context. But I don't how he comes to this conclusion, why can't all words mean an exact thing?

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why can't all words mean an exact thing?

The most concise answer you are going to find is in Section 293: the famous "beetle in a box" thought experiment.

If I say of myself that it is only from my own case that I know what the word "pain" means - must I not say the same of other people too? And how can I generalize the one case so irresponsibly?

Now someone tells me that he knows what pain is only from his own case! --Suppose everyone had a box with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what a beetle is only by looking at his beetle. --Here it would be quite possible for everyone to have something different in his box. One might even imagine such a thing constantly changing. --But suppose the word "beetle" had a use in these people's language? --If so it would not be used as the name of a thing. The thing in the box has no place in the language-game at all; not even as a something: for the box might even be empty. --No, one can 'divide through' by the thing in the box; it cancels out, whatever it is.

That is to say: if we construe the grammar of the expression of sensation on the model of 'object and designation' the object drops out of consideration as irrelevant.

As for the example of mathematics specifically, you'll want to look at the argument that begins in section 143, and goes on right up to the Beetle Box argument mentioned above. In short, mathematics is based upon "rule following", and as Wittgenstein shows (in Section 201), "No course of action could be determined by a rule, because every course of action can be made out to accord with the rule."

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    Excellent concise summary (with original source references!).
    – Mitch
    Dec 16, 2011 at 14:43
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    -1 I'm sorry but this is a complete misreading of Wittgenstein. He is not talking about language in those passages, but about the limitations of philosophical parlance. The beetle in the box is not an argument against semantic realism, but against philosophy's conceit of thinking it can talk meaningfully about essentially private states. The beetle in the box is specifically targeted against the philosophy of mind, which eg seeks to clarify 'pain' or (the exeprience of) 'red'.For W of course words can 'mean exact things' - just not those words used by philosophy. Always remember: Witt attacks
    – Chuck
    Dec 17, 2011 at 1:10
  • (cont.) philosophical language, not language in general.
    – Chuck
    Dec 17, 2011 at 1:11
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    And the rule-following arguments are in no way intended to show the indeterminacy of language. What Wittgenstein is attacking is the philosophical temptation to think of processes in terms of how well we grasp some interpretation or other of them. (See my answer here: philosophy.stackexchange.com/questions/1639/…) So for math what he would say, at most, is that it is pointless for philosophy to ask questions like 'Is mathematics a precise language?' Because to do so would involve thinking in terms of 'object and interpretation.'
    – Chuck
    Dec 17, 2011 at 1:18
  • @Chuch: I think you are missing the thrust of my argument, as we are actually in agreement. The question above is about the creation of "a perfect language", with mathematics as an example-- my answer is that Wittgenstein uses several methods (including the Beetle box and rule-following arguments) to demonstrate that such a perfect language is an impossible philosophical conceit. Naturally, that doesn't stop us from actually using language, or doing mathematics-- but that is not what the question was about.
    – Michael Dorfman
    Dec 17, 2011 at 12:40
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Yes, even in mathematics one could psychologically 'game' the system, where everyone is just agreeing that they have a beetle in their box, or the concept '5+7 = 12'. Except one can talk about the beetle, and one must agree about the properties of the beetle. One can always lie, but adversarily eventually with enough questions one would be found out (and with mathematical questions, not many are needed before one is found out.

There is a simple practical counter-... not exactly counter argument but just a -defense- of your position against Wittgenstein, which is that mathematical and rule based thinking, is, of any kind of rationally oriented thinking, the -most- perfect (W is just quibbling over how close to perfect). At a certain point of mathematical thinking (mostly in foundations/FOM) there is a sort of 'theological' character where one relies on a kind of faith or blind assumption. But after the barest moment of doubt, mathematical work gets along perfectly fine as a rule based system because the rules are so easy to make explicit (uh...easy-er to make explicit).

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The fact that mathematics is not a perfect language was pointed first by Hume and then Russell about induction and this is in line with the answer of @Michael. Even for the construction of numbers, axiomatic does not prevent you from the final instantiation of your general abstract model in the reality. At some point, this instantiation is made possible by your experience of the reality, this brings you somehow to induction where the finiteness of your experience avoid any “perfection” (in statistics, this is called “no free lunch”).

I think this is all already contained in Heraclite’s words (and related fragments):

Mais bien que le Logos soit commun

La plupart vivent comme avec une pensée en propre.

(sorry I only have the french translation From Jean Paul Dumont, in english that could be But even if the Logos is common to everyone Most are like having their own thought. (Note that this Fragment is reported by Sextus Empiricus in “Against Mathematics” … :) )

At the end, this depends on the meaning you give to “perfect” in perfect language. I found the word perfect a bit of a problem here. I guess you make the distinction between a perfect language and a closed language. You might be able to create closed languages Boole algebra is closed. Is it what you mean by “perfect”.

Coming back to Wittgenstein, have you read “on certainty” ? Also “Différences et repetitions” from Deleuze, would fit here...

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    D+R is a fantastic suggestion; in passing, while not nearly as focused on maths, his Logic of Sense might be helpful as well
    – Joseph Weissman ♦
    Dec 16, 2011 at 15:59

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