Not Plus
A favorite line of thought of mine that Wittgenstein took was to ask a question about a hypothetical form of negation. In normal English, the use of two 'not's, that is, two negations, serve as an affirmation: essentially the second negation undoes the first. We find this, for example, in logic, where ¬¬P↔P. But suppose, he says, we have another word that signifies negation, just like 'not', except in cases where it is repeated, in which case it serves to emphasize the original negation rather than turn it to affirmation†. No logical symbol suffices for emphasis, but I trust the sense of the following is clear: suppose þ is this new form of negation, then þþP↔þP!, and þþþP↔þP!!, and so on. The question he asks is: when we encounter a single use of either symbol, do they mean the same thing?
Time and time again in this forum I've encountered people who insist on the, ah, objectivity of mathematical meaning. It always puzzles me. I'm not sure if it comes from the way they learned math, or their affection for its "certainty," or that someone once told them smart folks said math is known a priori, or what. I've never come to the root of it, honestly. But neither have I been able to clearly express why I find it so puzzling.
I am a student of electronics, and in digital circuits we do a great deal with two-valued Boolean algebra. The reason for this is usually two-fold: first, to find the logical expression with the least number of terms, and second, to use this to create the circuit using the least number of inexpensive components. The first half is easy to understand. The second, however, is not, so I shall explain it. In digital logic (this two-valued Boolean algebra), all logical operations (these being AND, OR, XOR, XNOR, and NOT) can be performed with some combination of NAND gates. Component-wise, NAND gates are the simplest to impliment so they are cheaper. Secondly, you don't just bring in a single gate: chips usually come with four or more gates on board, so once you've used one NAND, you might as well use the other three, if you can. This means that the "simplest logical expression" is not just a matter of the least number of terms.
Logically, NAND is a "more complicated" operation in that it requires two logical operators: ¬ and ∧. But in actually building a circuit, this quaint idea is not carried through: the more complicated expression is often the simpler one.
If someone said to me, "Which is the simpler expression?" and gave me two logical expressions, I would not be able to answer them. I could simply say, "This has the least number of input terms, if that is what you mean by 'simple'." Reducing a complex expression to a simpler one is not an uncommon task (in logic, or in mathematics in general), but just because one does this with logical and mathematical expressions does not somehow grant it any weight: "simple" can mean different things. Just because we are dealing with math or logic does not grant us anything extra.
Lest one thinks this is a huge diversion, the point comes clear here. For the operation of OR gates, we use the symbol '+'. In evaluating the expected output of a logical OR operation, we might say, "1+1=1." Of course, in the algebra we learn in or before high school, "1+1=2" is the proper statement.
So I will ask you, what does '+' mean in the statement 0+1=1?
My answer is this: whatever '+' means, it is prescribed by its use. We are not representing some platonic ideal with the symbol. The knowledge of what '+' isn't there for us independent of experience. We were taught its use. In some cases, some of us were only taught one use, but others of us know two, or even more, uses for this symbol, all of which make sense. And I don't mean we learn the use of the symbol which represents some a priori concept, law, or rule. I mean we learn what it means by experience. The symbol doesn't attach to some concept, or stand in place of one; it isn't the feeble, limited representation of an epistemologically certain ideal. It is how it is used in whatever case.
That is how, too, I would answer Wittgenstein's question about negation. When the symbols stand alone-that is, when the context does not demand we differentiate-then they "mean the same thing" because they are used the same way. One can take the role of a child and ask, "What am I to do with this '+' in this case?" -And that is what '+' means. "What am I to do with this negation in this sentence?" -And that is what it means. There's nothing to get behind. It is right there for us, prescribed. The role for an expression with this symbol (or this curse word, or this exclamation, or etc.) was laid out for us and no one "really means" anything else by it but what was prescribed-unless, of course, someone prescribes yet another use (in the terminology of Philosophical Investigations: unless someone prepares a role for it in another language-game).
If we have any certainty about a symbol, it is related to mankind's prescription about its use, be that a word for negation, a symbol for addition, or the number one.
Someone once said to me, roughly, "Of course a person can say that a symbol "means" whatever they want and use it that way, but if they are talking about the number one then we are talking about the same thing." First, of course a person can prescribe a use for such symbols because the entire life of such symbols is hinged on such prescription. The illusion this aporetic objector of mine has is that one can somehow strip the symbol from the meaning to lay the meaning bare. Second is merely the question, "The same thing? What thing?" To me that rhetorical question berates the notion that a symbol attaches to-clothes, as it were-a meaning, and that if we only stripped away these things we'd find ourselves in a better place. But I've not found others see that point in the question. I almost imagine them saying, "Of course the same thing, one!" (Wittgenstein remarking: perhaps while holding up one finger...)
† It is not especially strange hypothetical in general. Many languages allow for double negation to effect a sense of emphasis rather than turn it to affirmation.
Time and time again in this forum I've encountered people who insist on the, ah, objectivity of mathematical meaning. It always puzzles me. I'm not sure if it comes from the way they learned math, or their affection for its "certainty," or that someone once told them smart folks said math is known a priori, or what. I've never come to the root of it, honestly. But neither have I been able to clearly express why I find it so puzzling.
I am a student of electronics, and in digital circuits we do a great deal with two-valued Boolean algebra. The reason for this is usually two-fold: first, to find the logical expression with the least number of terms, and second, to use this to create the circuit using the least number of inexpensive components. The first half is easy to understand. The second, however, is not, so I shall explain it. In digital logic (this two-valued Boolean algebra), all logical operations (these being AND, OR, XOR, XNOR, and NOT) can be performed with some combination of NAND gates. Component-wise, NAND gates are the simplest to impliment so they are cheaper. Secondly, you don't just bring in a single gate: chips usually come with four or more gates on board, so once you've used one NAND, you might as well use the other three, if you can. This means that the "simplest logical expression" is not just a matter of the least number of terms.
Logically, NAND is a "more complicated" operation in that it requires two logical operators: ¬ and ∧. But in actually building a circuit, this quaint idea is not carried through: the more complicated expression is often the simpler one.
If someone said to me, "Which is the simpler expression?" and gave me two logical expressions, I would not be able to answer them. I could simply say, "This has the least number of input terms, if that is what you mean by 'simple'." Reducing a complex expression to a simpler one is not an uncommon task (in logic, or in mathematics in general), but just because one does this with logical and mathematical expressions does not somehow grant it any weight: "simple" can mean different things. Just because we are dealing with math or logic does not grant us anything extra.
Lest one thinks this is a huge diversion, the point comes clear here. For the operation of OR gates, we use the symbol '+'. In evaluating the expected output of a logical OR operation, we might say, "1+1=1." Of course, in the algebra we learn in or before high school, "1+1=2" is the proper statement.
So I will ask you, what does '+' mean in the statement 0+1=1?
My answer is this: whatever '+' means, it is prescribed by its use. We are not representing some platonic ideal with the symbol. The knowledge of what '+' isn't there for us independent of experience. We were taught its use. In some cases, some of us were only taught one use, but others of us know two, or even more, uses for this symbol, all of which make sense. And I don't mean we learn the use of the symbol which represents some a priori concept, law, or rule. I mean we learn what it means by experience. The symbol doesn't attach to some concept, or stand in place of one; it isn't the feeble, limited representation of an epistemologically certain ideal. It is how it is used in whatever case.
That is how, too, I would answer Wittgenstein's question about negation. When the symbols stand alone-that is, when the context does not demand we differentiate-then they "mean the same thing" because they are used the same way. One can take the role of a child and ask, "What am I to do with this '+' in this case?" -And that is what '+' means. "What am I to do with this negation in this sentence?" -And that is what it means. There's nothing to get behind. It is right there for us, prescribed. The role for an expression with this symbol (or this curse word, or this exclamation, or etc.) was laid out for us and no one "really means" anything else by it but what was prescribed-unless, of course, someone prescribes yet another use (in the terminology of Philosophical Investigations: unless someone prepares a role for it in another language-game).
If we have any certainty about a symbol, it is related to mankind's prescription about its use, be that a word for negation, a symbol for addition, or the number one.
Someone once said to me, roughly, "Of course a person can say that a symbol "means" whatever they want and use it that way, but if they are talking about the number one then we are talking about the same thing." First, of course a person can prescribe a use for such symbols because the entire life of such symbols is hinged on such prescription. The illusion this aporetic objector of mine has is that one can somehow strip the symbol from the meaning to lay the meaning bare. Second is merely the question, "The same thing? What thing?" To me that rhetorical question berates the notion that a symbol attaches to-clothes, as it were-a meaning, and that if we only stripped away these things we'd find ourselves in a better place. But I've not found others see that point in the question. I almost imagine them saying, "Of course the same thing, one!" (Wittgenstein remarking: perhaps while holding up one finger...)
† It is not especially strange hypothetical in general. Many languages allow for double negation to effect a sense of emphasis rather than turn it to affirmation.