U1 have been hard at work.

(Concept and the vast majority of the creation Eleanor’s. Snowman Savannah’s; we all chipped in with the orthography…)

U1 have been hard at work.

(Concept and the vast majority of the creation Eleanor’s. Snowman Savannah’s; we all chipped in with the orthography…)

Genevieve took her final steps today.

What a great car:

- Cost next to nothing to buy or run, even in the days when I knew next to nothing about cars and didn’t know how to buy them REALLY cheap;
- Never ever ever failed to start. Still ran, even when she did once break;
- Even better than ever after new cylinder head last winter.

Sadly, the expenses of life left me with no choice, and keeping two cars became indefensible when I no longer had to drive to work every day.

It is a ludicrous and dishonest road tax system that penalises the truly green motorists who reduce environmental damage by keeping old cars going, while exempting the filthy new so-called eco-diesels/diesel hybrids, the carbon costs of whose manufacture alone equate to decades of running an older motor instead, and which pump far viler stuff into the environment to kill us and our children.

Yet again, though guilty in the past of promoting diseasel even more vigorously than the UK, the French lead the way.

But I still feel like a murderer.

Either way, F1 have definitely earned a reward. How about a Volvo? You don’t see one of these every day. It’s a 544 – spotted in St Giles’ at lunchtime today.

(It was only yards from this spot that I was approached last year by a complete stranger who offered to buy Phoenix. There must be something about the place.)

What a wonderful session with F1. 70 minutes of full-on argument from which, despite some pessimistic cries of despair, I think genuine progress was made.

Here’s an attempt to summarise where we got to. Please correct/dispute as seems fit.

There were some new matters on which I think we had broad agreement. Namely:

1. Neither simple terms like ‘12,051’ (or ‘twelve thousand and fifty-one’ – it makes no difference whether you use symbols or words), nor complex mathematical expressions like ’13 x 927′, refer to anything. There aren’t real entities called numbers, in addition to the mathematical expressions, and to which those mathematical expressions point. In this respect, maths is different from many expressions in ordinary language, such as ‘Desmond Morris’ or ‘the author of The Naked Ape’. For both of those expressions refer to an individual person. There is a reality beyond the language, to which the terms in the language point. And that is not the case with maths. (This is, at least, a common-enough and definitely arguable view about the ‘reality’ of numbers).

2. ‘The waitress is beautiful’ means the same thing as ‘la serveuse est belle’.

3. The “mathematicians’ side” of the debate added (and I now think they are (at least arguably) right): the expressions ’13 x 927′ and ‘12,051’ (and all mathematical expressions that are mathematically equal to these) have the same meaning.

4. I noted also (and I don’t think it is disputed) that you can know what ‘the waitress is beautiful’ means without knowing any French. It would be absurd to deny it. And in the same way you can know what ’13 x 927′ means without knowing that 13 x 927 = 12,051.

So it is possible to know what an expression means, even if you cannot come up with all expressions that are equivalent to it in meaning.

I was therefore wrong to suggest that, if you claim that ‘12,051’ is equivalent in meaning to ’13 x 927′, then to count as knowing the meaning of ’13 x 927′ you would have to be able to answer, instantly, ‘12,051’ when asked what ’13 x 927′ was when expressed as an integer. You can know what an expression means – both in maths and in ordinary language – without knowing every expression that has the same meaning.

So much for the points of agreement.

BUT: we mustn’t lose sight of the original issue. What we were disputing was whether the mathematical example used in this post served to make the relevant point against Meno’s paradox – the point that the target of inquiry could be known under one description but not under another; and that inquiry was thus the legitimate and purposeful exercise of getting to know the target under a different description from the one originally known.

Now: given the first matter of broad agreement above – that mathematical expressions don’t refer, it will be true that there is no entity to which each of the expressions ’13 x 927′ and ‘12,051’ refer. So the inquiry – the calculation – cannot be a matter of coming to know such an entity under a new description – knowing it under its ‘integer-name’ rather than under a more complex description involving the multiplication sign. In that respect, the case differs from the Desmond Morris example.

But the (still) undeniable fact remains – I think – that you do learn something new.

What you learn is, I am now inclined to say, new knowledge about the expression with which you started: before you did the calculation, you did not know that the expression ’13 x 927′ was equal to 12,051; afterwards, you did know this fact. The target of the inquiry is not a (non-existent) number to which both expressions refer; it is the expression that appears in the question.

(It is, by the way, a complete irrelevance that there is a tiny minority of rather unusual people – the so-called ‘idiots savants’ – who don’t suffer from such ignorance. The fact is that the majority of us learn something new. Yes, this is because of a human weakness – ordinary people can’t instantly spot all the expressions mathematically-equivalent to a given expression. But that observation doesn’t make the fact that most of us learn something new go away.)

When the maths teacher sets the question ’13 x 927 = ?’, (s)he wants the student to provide a single integer, as opposed to a complex expression, as the answer. That’s just implicit in the type of question set. If (s)he wanted it in a different form (i.e. a different but mathematically equivalent expression), (s)he would have said so.

The student – unless (s)he is an idiot savant – does not initially know the required answer. That is not to say that (s)he doesn’t know the meaning of ’13 x 927′, nor that (s)he doesn’t know the meaning of ‘12,051’. (S)he knows the meaning of both those expressions; (s)he just doesn’t know that they mean the same; that is because (s)he hasn’t done the work that will show her/him that they are mathematically equal. (So what is learned is a fact about meaning: it is the fact that this and that expression have the same meaning. It is like learning that ‘la serveuse est belle’ means the same thing as ‘the waitress is beautiful’. But from the fact that it is a fact about the meaning of these expressions it doesn’t follow that you didn’t already know what either expression meant.)

So the student doesn’t know the expression ’13 x 927′ under the description that the teacher wants; the inquiry (doing the long multiplication/pressing the buttons on the calculator (if you can call that ‘inquiry’)) is a matter of coming to see that that expression is mathematically equal to the expression ‘12,051’.

In which case the example does still work, and in the same way, against Meno. But I concede that, given the meta-mathematical claim that numbers aren’t real entities to which number expressions refer, the target of the inquiry has to be re-characterised as not the reference of the expressions, but as the expression itself that appears in the question.

Will that satisfy anyone, or will it merely generate further dispute, I wonder?

While I am reminiscing, let me report that on Friday evening I was in the same supermarket as mentioned in this post.

As I approached the till, I realised that the person who would be serving me was familiar… so of course I started praying that it wouldn’t be a recognisable date…

Fat chance.

“Sixteen-oh-three”, she says.

“Too easy: death of Elizabeth the First.”

“I don’t know much history.”

“Neither do I. But I think you remember that nothing happened in 1001…?”

“Oh yes.”

My mind has been slightly exercised – thank you – by the objection raised by some members of F1 to an example that I offered in support of what seems to me to be the obvious solution to Meno’s paradox. Meno’s paradox, recall, is the claim that inquiry must be either pointless or impossible; for:

- If you don’t already know what it is that you are inquiring into, you won’t recognise it even if you come across it.

But

- If you do already know what you’re inquiring into, then there is no need for the inquiry in the first place.

The obvious solution, of course, is to say that in the relevant kind of inquiry we know the target under one description, but not under some other description. So if I ask you to find out who the the author of *The Naked Ape* is, you do already, in virtue of the way in which I have set up the project, know the target of your inquiry under this description: you know that the person you are seeking is the author of that book; but you don’t know him under every description: you don’t, I am assuming, know his name. The inquiry is therefore not pointless: your aim is to find out something else about the target; and the inquiry is possible, because since you do know something about the target, you do have the resources to recognise it as the answer to your inquiry when you come across it.

The issue arose whether asking ‘What is 13 x 927?’ will do equally well as illustrating this solution to the supposed paradox.

My claim is that, when you are asked this question, you know the target of your inquiry under this description: you know that the thing you are seeking is a number that is the product of 13 and 927; but you don’t know it under every description: you don’t, I am assuming, know the name of the relevant integer.

People complained, however, that the arithmetical case is different: roughly, the objection as I understood it was that since all numbers are defined by reference to their various mathematical relations to all other numbers, the very meaning of the expression “13 x 927″ is, or includes, the fact that it is 12,051. Accordingly, people seemed to be saying, there isn’t anything that you don’t already know about the target of the inquiry, by the time I have set the question. That the answer is 12,051 isn’t new knowledge, but something that was already given in the question set.

Now:

My intuition is that this must be wrong: it seems to me clear and undeniable that we all learned something new the first time we did the calculation and found that the right answer is 12,051. The question is how to defend that view, given that I am inclined to accept the claim that all numbers are defined by reference to their various mathematical relations to all other numbers.

Proposal: As well as the familiar and orthodox distinction between meaning and reference, illustrated by (e.g.)

- ‘The evening star’ and ‘the morning star’ mean different things, though they both refer to the same entity.

We also need to distinguish between meaning and ‘definition’ – at least for these pesky numbers.

With such a distinction, we could say that every number is defined by reference to mathematical relations to all other numbers; but it doesn’t follow that the term, or word, ‘seven’ means anything like ‘5 less than 12′ – not even in part.

‘Definition’ here may not be the right or most helpful word; what I am really pointing to is that the identity conditions of each number – what it takes for it to be the number that it is – may well include relations to all the other numbers. But there is no reason to suppose that all of those identity conditions have entered into the meaning of the name of each individual number. Indeed, there are good reasons for saying that they haven’t:

- it lets us say what is intuitively correct – namely that we do learn something new when we do the complicated arithmetic and find the solution to the problem set.
- it lets us say that small children know the meaning of terms like ‘seven’, even though they don’t know much maths.
- it lets us say that we know the meaning of terms like ‘seven’, even though we have finite minds and cannot possibly know all of the infinitely-many relations to the infinitely-many other numbers.
- (It also, just by the way, makes sense of the Kantian claim that ‘7 + 5 = 12′ is synthetic. But see Rule 4.)

Here’s the link to the excellent film mentioned today – startlingly similar in key elements to the one described in this post:

- hero has lost his wife
- hero’s goodness is revealed by his job, which involves concern for the young (teacher in the one, paediatrician in the other)
- hero’s goodness is revealed, much more tellingly, by his possession of a proper old Volvo estate (745, 945)
- both films use two different Volvos, hoping that the audience won’t notice.

Spooky.

(You get a glimpse of one of the Volvos in the opening seconds of the trailer, but blink and you miss it.)

See also here.

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