§189
Wittgenstein does not want to deny that, "the steps are determined by the formula..." makes sense. He just asks how the expression is used. His method, I assume, is to get rid of conceptual confusion by drawing attention to uses of the expression which make sense.
Training/education is mentioned here. Would it be correct to say that the right way to continue the series is the way that people with the right sort of education would continue? Presumably not - people educated in mathematics might make mistakes. But there is the connection with mastery of a technique here - gaining the appropriate abilities.
Wittgenstein suggests that we might say that "for these people [trained to take the same step at the same point] the order '+3' completely determines every step from one number to the next". But this doesn't sound right. Surely it would be correct for someone to write '1006' after '1003' even if they hadn't shared in the training of the relevant group. Or is Wittgenstein's point that another group might have different concepts similar to our number concepts?
"'y = x(squared)' determines a number y for a given value of x' is a grammatical truth - so it's unclear what to make of the question 'is y = x(squared) a formula which determines y for a given x?'
What was incorrect in the interlocutor's view? - It is a mistake to think that the formula makes certain steps correct (as if there is some intermediary between formula and application) a given step is correct - in accord with the rule.
Monday, 20 May 2013
§188
At the end of §187 Wittgenstein asked what was wrong with the interlocutor's (A's) idea. What was the interlocutor's idea? - That in meaning the order A had already taken all the various steps- "In meaning it your mind, as it were, flew ahead and took all the steps before you physically arrived at this or that one" - meaning something anticipates reality.
In §188, as far as I can tell, Wittgenstein does not say what was wrong with the interlocutor's idea - he just spells out what the idea was.
In 'Accord with a rule' - in the analytical commentary vol.2 - Hacker compares the problem here with 'the mystery of negation'. The problem is that (iii) seems to conflict with (i) and (ii) in the following triad of propositions:
(i) What a true proposition depicts is what is the case.
(ii) What a false proposition depicts is not what is the case.
(iii) What a proposition depicts is the same, no matter whether it is true or false.
In the Tractatus Wittgenstein had 'solved' this mystery by saying that what a proposition depicts is a state of affairs - which is a possibility, not an actuality. States of affairs mediate between propositions and facts.
Similarly in the case of following a rule meaning the steps in the extension of the series is mediate between the rule and the applications of it - it is what connects the two (this is the confused view of the interlocutor).
At the end of §187 Wittgenstein asked what was wrong with the interlocutor's (A's) idea. What was the interlocutor's idea? - That in meaning the order A had already taken all the various steps- "In meaning it your mind, as it were, flew ahead and took all the steps before you physically arrived at this or that one" - meaning something anticipates reality.
In §188, as far as I can tell, Wittgenstein does not say what was wrong with the interlocutor's idea - he just spells out what the idea was.
In 'Accord with a rule' - in the analytical commentary vol.2 - Hacker compares the problem here with 'the mystery of negation'. The problem is that (iii) seems to conflict with (i) and (ii) in the following triad of propositions:
(i) What a true proposition depicts is what is the case.
(ii) What a false proposition depicts is not what is the case.
(iii) What a proposition depicts is the same, no matter whether it is true or false.
In the Tractatus Wittgenstein had 'solved' this mystery by saying that what a proposition depicts is a state of affairs - which is a possibility, not an actuality. States of affairs mediate between propositions and facts.
Similarly in the case of following a rule meaning the steps in the extension of the series is mediate between the rule and the applications of it - it is what connects the two (this is the confused view of the interlocutor).
§187
Wittgenstein agrees with the interlocutor that it makes sense to say that, "I already knew that B should write 1002 after 1000 when I gave the order". Philosophical/conceptual confusion can arise if this is misconstrued, however.
To say that you meant for B to write '1002' after '1000' when you gave the order '+2' does not mean that you thought of the step when you gave the order.
The interlocutor says that, "if I had been asked what number he [B] should write after 1000 I would have replied '1002'". Wittgenstein says that he doesn't doubt this. Does this mean that meaning something means having certain dispositions?
--Philosophical problem: would it mean that you'd have to have had an infinity of dispositions? - If you were asked what number B should write after 5,016 is it clear that you'd have replied '5,018'? And 10,314 after 10,312 etc. etc.
Wittgenstein wants to get away from the confused idea that knowing something is an event that occurred simultaneously with giving the order (the interlocutor says 'I..knew, at the time when I gave the order...'). Wittgenstein has already made the point, in §150, that the grammar of the word 'know' is closely related to 'can' and 'is able to'.
When Wittgenstein says that it would be correct for A to say, "if I had then been asked what number he should write after 1000, I would have replied '1002'" he says so because he does not doubt that A has mastered basic mathematics - that A is able to add 2 to any number (not because A thought of this case at the time of giving the order).
Wittgenstein agrees with the interlocutor that it makes sense to say that, "I already knew that B should write 1002 after 1000 when I gave the order". Philosophical/conceptual confusion can arise if this is misconstrued, however.
To say that you meant for B to write '1002' after '1000' when you gave the order '+2' does not mean that you thought of the step when you gave the order.
The interlocutor says that, "if I had been asked what number he [B] should write after 1000 I would have replied '1002'". Wittgenstein says that he doesn't doubt this. Does this mean that meaning something means having certain dispositions?
--Philosophical problem: would it mean that you'd have to have had an infinity of dispositions? - If you were asked what number B should write after 5,016 is it clear that you'd have replied '5,018'? And 10,314 after 10,312 etc. etc.
Wittgenstein wants to get away from the confused idea that knowing something is an event that occurred simultaneously with giving the order (the interlocutor says 'I..knew, at the time when I gave the order...'). Wittgenstein has already made the point, in §150, that the grammar of the word 'know' is closely related to 'can' and 'is able to'.
When Wittgenstein says that it would be correct for A to say, "if I had then been asked what number he should write after 1000, I would have replied '1002'" he says so because he does not doubt that A has mastered basic mathematics - that A is able to add 2 to any number (not because A thought of this case at the time of giving the order).
Friday, 17 May 2013
§186
In §186 Wittgenstein raises the possibility (in the voice of an interlocutor) that a new intuition is needed at each step to carry out the order correctly. I'm not clear about what is meant by 'intuition' here. In §186 Wittgenstein uses 'intuition' as if it is equivalent to 'insight' (according to the Hacker/Schulte translation). Later on, in §213 Wittgenstein talks about intuition as an 'inner voice'. My dictionary says 'quick and ready insight' or 'hunch'. Is intuition also used to mean something like knowing/feeling sure but without evidence?
If intuition is an 'inner voice' is it your voice? - Presumably there are similar sceptical worries about what you mean by the words which make up your thoughts. - If there are sceptical worries about rule following then there are presumably also sceptical worries about what we mean by our words/what our words mean (because an explanation of the meaning of a word plays the role of a rule for the use of that word).
In §213 Wittgenstein scotches the suggestion that an intuition is needed at each step: "If intuition is an inner voice - how do I know how I am to follow it? And how do I know that it doesn't mislead me? For if it can guide me right, it can also guide me wrong."
In §186 it seems that Wittgenstein also wants to reject the possibility that the correct way to go on is the way that A intended:
(1) ...because if A meant for B to write '1002' after '1000' then A presumably also meant for 'B' to write '15,544' after '15,542' and '89,012,018' after '89,012,016' - but this would mean that A intended an infinity of steps at once, which is impossible.
(2) If A were to respond by saying that they didn't mean an infinity of sentences but rather they meant "that B should write the next but one number after every number that he wrote; and from this stage by stage, all those sentences follow." then it seems reasonable to respond that this is not satisfactory either - because if there are worries about '+2' then there are also worries about the sentence in italics.
This does sound as if Wittgenstein is setting up a sceptical problem. But it could be that Wittgenstein is presenting the philosophical problem in a form that those with the relevant vexations/confusions would recognise before he goes on to dissolve the problem - lay out the conceptual terrain in a surveyable representation.
In §186 Wittgenstein raises the possibility (in the voice of an interlocutor) that a new intuition is needed at each step to carry out the order correctly. I'm not clear about what is meant by 'intuition' here. In §186 Wittgenstein uses 'intuition' as if it is equivalent to 'insight' (according to the Hacker/Schulte translation). Later on, in §213 Wittgenstein talks about intuition as an 'inner voice'. My dictionary says 'quick and ready insight' or 'hunch'. Is intuition also used to mean something like knowing/feeling sure but without evidence?
If intuition is an 'inner voice' is it your voice? - Presumably there are similar sceptical worries about what you mean by the words which make up your thoughts. - If there are sceptical worries about rule following then there are presumably also sceptical worries about what we mean by our words/what our words mean (because an explanation of the meaning of a word plays the role of a rule for the use of that word).
In §213 Wittgenstein scotches the suggestion that an intuition is needed at each step: "If intuition is an inner voice - how do I know how I am to follow it? And how do I know that it doesn't mislead me? For if it can guide me right, it can also guide me wrong."
In §186 it seems that Wittgenstein also wants to reject the possibility that the correct way to go on is the way that A intended:
(1) ...because if A meant for B to write '1002' after '1000' then A presumably also meant for 'B' to write '15,544' after '15,542' and '89,012,018' after '89,012,016' - but this would mean that A intended an infinity of steps at once, which is impossible.
(2) If A were to respond by saying that they didn't mean an infinity of sentences but rather they meant "that B should write the next but one number after every number that he wrote; and from this stage by stage, all those sentences follow." then it seems reasonable to respond that this is not satisfactory either - because if there are worries about '+2' then there are also worries about the sentence in italics.
This does sound as if Wittgenstein is setting up a sceptical problem. But it could be that Wittgenstein is presenting the philosophical problem in a form that those with the relevant vexations/confusions would recognise before he goes on to dissolve the problem - lay out the conceptual terrain in a surveyable representation.
§185
In §185 Wittgenstein returns to the example from §143 - the language game of A issuing orders to B and B then writing down a series/continuing a series according to a rule.
In sections §§143-184 Wittgenstein clarified the grammar of 'understand'. He raised the question: how can it be that what we understand/grasp in an instant (the meaning of an expression) is its use (which is extended in time)? - This could be understood as a rhetorical question meant as an attack on the claim that meaning is use. Wittgenstein said in those passages that it was a mistake to think that understanding an expression was a mental state from which correct use flowed. Understanding is not a mental state. Unlike mental states it lacks genuine duration.
Now, in §185 he commences a discussion concerning what counts as accord with a rule and what counts as following a rule. His discussing of rule-following parallels his discussion of meaning and understanding from earlier. An explanation of meaning is a kind of rule (it sets a standard of correctness) and a criterion for understanding an expression is correct application. Being able to give a correct explanation of the meaning of a word is also a criterion for understanding it. However, it seems clear that someone might satisfy one criterion but not the other (they might give a correct explanation of meaning and yet go on to use the word in question incorrectly). What are we to say about this?
Similar to the kind of case just mentioned is someone stating a formula (a kind of rule - parallel to an explanation of meaning - both give a standard of correctness and can be used to justify/criticise actions) and someone applying the formula (which parallels someone using a word correctly or incorrectly). So his discussion of rule-following bears upon his discussion of meaning and use.
In §185 the pupil (B) is asked to continue the series '+2' from 1000 upwards. B has previously demonstrated the ability to add 2 to numbers below 1000. B continues the series '1000, 1004, 1008, 1012'. When challenged about this ('you should have added 2!') B says 'I did add 2. - I went on in the same way.' What can we say to B to convince them that they did not go on in the same way? It seems that repeating what has already been said in the course of instructing B would be pointless. They think that they've gone on in the same way despite having had that training. Does the rule 'add 2' determine a correct way to go on in every instance? - It seems that Wittgenstein wants to say 'yes'. 'x + 2 = y' would count as a formula which determines a number, y, for a given value of x according to §189. What Wittgenstein wants to do is to clear up any confusion surrounding what this (the formula determining a value in each case) might amount to.
Some have found in these passages (§185 onwards) a sceptical argument leading to the paradoxical conclusion that "no course of action could be determined by a rule, because every course of action can be made out to accord with the rule" (§201) i.e. Kripke. - It could be that B interpreted '+2' to mean 'add 2 up to 1000, then add 4 up to 2000, and add 6 up to 3000 and so on'. Is there anything in the rule '+ 2' that rules out this way of interpreting it? A could respond by saying that the correct way to interpret the rule was the way that he (A) meant it. But did A, when giving the rule mean that B should write '1002' after '1000'? - This kind of sceptical concern is examined by Wittgenstein but it isn't clear that what Wittgenstein does is to raise sceptical worries and then proffer a sceptical solution. More likely, I think, is that Wittgenstein wants to expose nonsense and dissolve philosophical problems rather than offer a traditional/Humean solution to them.
Question: Would this be another way of posing the sceptical problem? -- Given that continuing the series '1002, 1004, 1006, 1008' and '1004, 1008, 1012, 1016' are both possible ways of interpreting '+2' how are we to decide which is the correct way of continuing the series?
In §185 Wittgenstein returns to the example from §143 - the language game of A issuing orders to B and B then writing down a series/continuing a series according to a rule.
In sections §§143-184 Wittgenstein clarified the grammar of 'understand'. He raised the question: how can it be that what we understand/grasp in an instant (the meaning of an expression) is its use (which is extended in time)? - This could be understood as a rhetorical question meant as an attack on the claim that meaning is use. Wittgenstein said in those passages that it was a mistake to think that understanding an expression was a mental state from which correct use flowed. Understanding is not a mental state. Unlike mental states it lacks genuine duration.
Now, in §185 he commences a discussion concerning what counts as accord with a rule and what counts as following a rule. His discussing of rule-following parallels his discussion of meaning and understanding from earlier. An explanation of meaning is a kind of rule (it sets a standard of correctness) and a criterion for understanding an expression is correct application. Being able to give a correct explanation of the meaning of a word is also a criterion for understanding it. However, it seems clear that someone might satisfy one criterion but not the other (they might give a correct explanation of meaning and yet go on to use the word in question incorrectly). What are we to say about this?
Similar to the kind of case just mentioned is someone stating a formula (a kind of rule - parallel to an explanation of meaning - both give a standard of correctness and can be used to justify/criticise actions) and someone applying the formula (which parallels someone using a word correctly or incorrectly). So his discussion of rule-following bears upon his discussion of meaning and use.
In §185 the pupil (B) is asked to continue the series '+2' from 1000 upwards. B has previously demonstrated the ability to add 2 to numbers below 1000. B continues the series '1000, 1004, 1008, 1012'. When challenged about this ('you should have added 2!') B says 'I did add 2. - I went on in the same way.' What can we say to B to convince them that they did not go on in the same way? It seems that repeating what has already been said in the course of instructing B would be pointless. They think that they've gone on in the same way despite having had that training. Does the rule 'add 2' determine a correct way to go on in every instance? - It seems that Wittgenstein wants to say 'yes'. 'x + 2 = y' would count as a formula which determines a number, y, for a given value of x according to §189. What Wittgenstein wants to do is to clear up any confusion surrounding what this (the formula determining a value in each case) might amount to.
Some have found in these passages (§185 onwards) a sceptical argument leading to the paradoxical conclusion that "no course of action could be determined by a rule, because every course of action can be made out to accord with the rule" (§201) i.e. Kripke. - It could be that B interpreted '+2' to mean 'add 2 up to 1000, then add 4 up to 2000, and add 6 up to 3000 and so on'. Is there anything in the rule '+ 2' that rules out this way of interpreting it? A could respond by saying that the correct way to interpret the rule was the way that he (A) meant it. But did A, when giving the rule mean that B should write '1002' after '1000'? - This kind of sceptical concern is examined by Wittgenstein but it isn't clear that what Wittgenstein does is to raise sceptical worries and then proffer a sceptical solution. More likely, I think, is that Wittgenstein wants to expose nonsense and dissolve philosophical problems rather than offer a traditional/Humean solution to them.
Question: Would this be another way of posing the sceptical problem? -- Given that continuing the series '1002, 1004, 1006, 1008' and '1004, 1008, 1012, 1016' are both possible ways of interpreting '+2' how are we to decide which is the correct way of continuing the series?
Sunday, 21 April 2013
§184
Wittgenstein is attacking the reservoir model of the mind found in William James. When you have forgotten a tune and then you are suddenly able to sing it does this mean that the tune was there in your mind in some sense?
- Presumably Wittgenstein wants to deny that the tune is present in the mind in the sense in which water is stored up in a reservoir and can be released at some point.
It certainly wasn't the case that you heard the whole tune in your mind in a flash.
It could be - as in the case of the person who declares 'now I know how to go on' when observing a series being written down - that you think you know the tune but find you don't. - Having a feeling that you know is no guarantee that you do.
Hacker: "The 'presence of the tune to the mind' is not the ground of, and explanation of, the certainty or the exercise of the ability".
Wittgenstein is attacking the reservoir model of the mind found in William James. When you have forgotten a tune and then you are suddenly able to sing it does this mean that the tune was there in your mind in some sense?
- Presumably Wittgenstein wants to deny that the tune is present in the mind in the sense in which water is stored up in a reservoir and can be released at some point.
It certainly wasn't the case that you heard the whole tune in your mind in a flash.
It could be - as in the case of the person who declares 'now I know how to go on' when observing a series being written down - that you think you know the tune but find you don't. - Having a feeling that you know is no guarantee that you do.
Hacker: "The 'presence of the tune to the mind' is not the ground of, and explanation of, the certainty or the exercise of the ability".
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