Part 2. Recent and Contemporary
11*. Frege, Wittgenstein and Platonism
When people call Frege a Platonist, one thing they have in mind is that he believed that numbers are objects. Some, ignorant of his writings more than I am, think that he believed that concepts were objects too. He is famous for having said ‘The concept horse is not a concept’; less famous for having thought the matter out further and written to the effect that e.g. ‘is what “horse” means’ is equivalent to ‘is a horse’. The essay didn’t get published, it just got a rejection slip.
I draw attention now to part of one of his greatest lectures, published as an article in 1891, ‘Funktion und Begriff’. He tells us there what was originally understood in mathematics by the word ‘function’. He remarks:
To this question one may well get as answer: ‘By a function of x was understood a mathematical expression containing x, a formula which includes the letter x.’ According to this explanation, for example, the expression
2.x3 + x
would be a function of x, and
2.23 +2
a function of 2. This answer cannot do, because in it form and content, sign and signified are not distinguished, an error which is often found in mathematical writings, even in celebrated authors, at the present day.
Referrring to his Grundlagen der Arithmetik, he says he has already pointed to the lack of viable formal theories in arithmetic. Signs with no content, and which aren’t supposed to have any nevertheless have properties attributed to them which can intelligibly attach only to the content of a sign. So here: a mere expression, the form for a content, cannot be the essence of the matter: only the content can be that. Now what is the content, the meaning of ‘2.23 + 2’? The same as of ‘18’ or of ‘3.6’.
All these expressions stand for—bedeuten—the same thing. Frege says he must set his face against
the opinion that for example 2 + 5 and 3 + 4, while equal, are not the same. In this opinion again, at bottom, resides that same confusion of form and content, of sign and signified. It’s like saying that the sweet smelling violet and viola odorata are different because the names sound different. Difference of designation by itself is insufficient to ground a difference in what is designated. Here the matter is less transparent only because the meaning of the numeral 7 isn’t anything perceptible to the senses. The present widespread inclination to acknowledge nothing as an object that cannot be perceived by the senses tempts in this case to take the numerals themselves for the numbers, for the proper objects of consideration, and then of course 7 and 2 + 5 would be different. But such a conception is untenable, because it is quite impossible to speak of any arithmetical properties of numbers without going back to the meaning [the Bedeutung] of the numerals. The property that 1, for example, yields itself when multiplied by itself, would be a mere fantasy: however far you carry a microscopic examination or chemical investigation, you could never discover this property in the innocent character that we call the numeral ‘one’. Perhaps a definition is spoken of—but no definition is creative in such a way as to be able to impart properties to a thing which it hasn’t got, apart from the one property of expressing and signifying what the definition introduces it as a sign of.
I am reminded of the passage where Frege, discussing the views of someone who said numbers were numerals, and who spoke of a number as getting constantly smaller, said ‘I see what you mean’ and printed a row of sevens, smaller and smaller across the page. I am reminded again of how Frege, in discussing people who explain numbers in terms of units, asks whether the units in 1 + 1 = 2 are the same or different—if the same, how can one say ‘one and one’? Can one say ‘The moon and the moon’? What is going on? And if they are different, what is it all about? And I am reminded of Socrates in the Phaedo saying he can’t understand how addition and division (both) are supposed to turn one into two. When I read the introduction to Frege’s Grundlagen my spirit bounded with the recognition of a brother to Socrates as so depicted by Plato.
Returning to ‘Funktion und Begriff’: a bit further on, Frege remarks that
(1 + 1) + (1 + 1) + (1 + 1) = 6, and so (1 + 1) is the number designated by 6:3.
The different expressions correspond to different conceptions and aspects but still always to the same thing. Otherwise the equation x2 = 4 would not just have the roots 2 and –2 but also 1+1 and innumerable others, distinct from one another though similar in a certain respect. In acknowledging that there are just two real roots one is rejecting the opinion that the equals sign means not a perfect co-incidence but only a partial agreement. Hanging on to this, we see that the expressions
‘2.13 + 1’
‘2.23 + 2’
‘2.43 + 4’
mean numbers, namely 3, 18 and 132. If a function were really the meaning of a mathematical expression a function would just be a number, and nothing new would have been attained for arithmetic. Of course in connection with the word ‘function’ one usually has in mind expressions in which a number is only indefinitely indicated by the letter x, e.g.
‘2.x3 + x’
but that doesn’t make any difference; for this whole expression too indicates a number only indefinitely, and whether I write down this expression or just x makes no essential difference.
All the same [he goes on], this fact about the indefinite indication by ‘x’ does lead us in the right direction. That indefinite indicator ‘x’ is called the argument of the function and we recognise the same function, but with distinct different arguments in
‘2.13 + 1’
‘2.43 + 4’
‘2.53 + 5’
Here the arguments are 1, 4 and 5. From this we can see that the proper essence of the function resides in what is common to these expressions, that is to say, in what is present in
‘2.x3 + x’
apart from the letter x—which we might also write as follows
‘2.( )3 + ( )’
Frege ‘wants to show that the argument doesn’t belong with the function as part of it but together with the function forms a complete whole’.
What, stopping here, can we learn from these considerations of Frege’s? Two things spring out. First, a similarity of form of expression does not necessarily betoken a similarity of what is expressed. Example: understood as expression of a function
‘2.x3 + x’
is quite different in meaning from
‘2.23 + 2’
Secondly, a function need not be signified by a sign which is to be found in every expression in which that function is expressed. The function was the same in
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2.13 + 1 |
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and |
2.43 + 4 |
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and |
2.53 + 5 |
not because of a particular sign designating the function, but because of what is common to those expressions. The fact that we might define φ as a function sign such that
φ(4) = Def. (2.43) + 4
or more generally
φ(x) = Def. (2.x3) + x
does not mean that anything false has been said in identifying this function as what, beyond (x), is common to expressions of that form. Supplied with arguments they give values of that function, which are numbers.
Let me sum up these two lessons thus:
the first is a principle of the possible non-identity of mode of signification of expressions which look and are significantly similar.
The second is a principle of identities being possibly visible only to the intellect.
I don’t mean that Frege started out from these principles as premises; rather he exhibited their truth in his reasoning. They will have been clear to him from when he wrote Begriffschrift.
Frege works through in the same article to the conclusion that a concept is a function from objects to truth-values. Here I am not especially concerned with this part of his work for my topic here is Wittgenstein. I have shown what I think there is in calling Frege a Platonist. This is for me the interest of the question: is Wittgenstein anti-Platonist and more particularly anti-essentialist? It might be thought that he was anti-Platonist from the story he told of asking Frege ‘Don’t you see any difficulty in calling numbers objects?’ To which Frege replied: ‘Sometimes I seem to see a difficulty, and then again I don’t see it’. Wittgenstein thought that this was a typical expression of a certain sort of thing that happens to one in doing philosophy. He was not excluding himself from this generalisation.
Frege reaches the account of a concept as a function from objects to truth-values via his addition of =, > and < to the functions +, -, exponentiation, ÷, etc. that he has been considering. The values of these functions are of course numerical; the values of the new functions are truth and falsehood, and so he introduces the terminology of truth-values (which comes from him) so he comes to functions, which he calls concepts, that often are not numerical at all.
His conception was rich and fruitful. Wittgenstein could not be supposed to object to the material I have been expounding up to this point about concepts; but here there is certainly a break. Frege’s conception involves regarding sentences as complex names of truth-values; it also involves a certain equality on the part of the two truth values: it is as if they were there to be designated independently of the construction of their designations. Whatever one may say about truth this appears highly objectionable about falsehood. Already in the Tractatus Logico-Philosophicus Wittgenstein had objected to both of these things—explicitly to Frege’s taking a sentence to be a complex name, but also in rejecting the notion that truth and falsehood are equally justified relations between signs and what is signified. Not that he is there assuming that the ‘signified’ in the case of propositions are the two truth-values; it is the equality of justification between truth and falsehood that he objects to. He objects to this whatever is thought to be signified by sentences.
However the part of Frege’s work in ‘Funktion und Begriff’ which I have identified as Platonist—that stands. So far as I know, the only places where Wittgenstein considers the expression itself to be what it expresses are aesthetic. A musical phrase, a bed of violets: such things may strongly give one the impression that they tell one something. What is it that they tell one? They tell one themselves, not something else.
At the beginning of the Philosophische Untersuchungen Wittgenstein describes a simple proceeding in the use of words. I ask a greengrocer for five red apples; I give him a slip of paper with these words so written on it. The shopkeeper
opens the drawer marked ‘apples’; then he looks at a colour table for the word ‘red’ and finds a colour sample by it; now he utters the series of cardinal numbers (he knows them by heart) up to the word ‘five’, and at each numeral he takes an apple out of the drawer whose colour is that of the colour sample. ... How does he know where to look up the word ‘red’ and what to do with the word ‘five’?—Well, I assume he acts as I have described. Explanations come to an end somewhere.—But what is the meaning of the word ‘five’?—No such thing came into the matter; only how the word ‘five’ gets used.†
I take it we should say: gets used in such and similar cases. If we are counting how many prime numbers come before twenty, one which we shall count is the number five. It has to be the number which is counted, for the property of being prime, Frege surely showed us, cannot belong to the numeral. Is 5 an object here, then, though no such object made its appearance in the transaction with the greengrocer?
I would say that Wittgenstein never—so far as I can tell—got that question really sorted out. In the Philosophische Grammatik he compares the feeling that Three is an object with the feeling or conviction that understanding is a process. But so far as I know he does not anywhere show that it is a grammatical illusion, as he does succeed in doing about understanding being a process.
The words ‘Five red apples’ are each of a different category. One learns the word ‘red’ as name of a colour, yet things one calls ‘red’ aren’t all in every sense the same colour. But in counting one doesn’t call anything ‘five’, say. If one were teaching a child to count, using oranges and pears as objects to be counted, and the child said ‘But last time you called that pear “one”, why are you calling an orange “one” now and that pear “three”?’ he would not have so much as begun to grasp the grammar, the technique of application, of numerals in counting. When one counts prime numbers one utters numerals and might be thought to be counting the numerals standing for prime numbers, as when one counts distinct printed words. But if one includes ‘nine’ one has made a mistake which can only be shown by considering the number: ‘9’, it will be said, ‘is not prime, it is 3×3’. Evidently one is not talking about the numeral, which cannot be said to have the property of being prime or of having factors.
‘Essence’, Wittgenstein said, ‘is expressed by grammar’. It’s been supposed that he meant something different by the word ‘grammar’ from what is ordinarily meant. Well, for what it is worth, I can testify that he claimed not to mean anything different. I heard him in class saying ‘What I mean by “grammar” is what you mean by it, what you heard lessons in at school’. Now at school the grammar we learned, if it was not Latin declensions and conjugations and what cases are taken by this or that preposition, and what genders various nouns are, and if it was not syntax, was an exercise called ‘parsing’. You learned what to call nouns and verbs and participles and conjunctions and adjectives and adverbs and to say what ‘governed’ what in a sentence, and you were given sentences which you had to parse—that is, say what was subject, what predicate, what direct or indirect object, what was an adjective attaching to what or used attributively, and so on. It was mildly interesting but it didn’t go very far. I knew a child, who had been taught that an adjective was a word describing what something is like; encountering a sentence which had the phrase ‘two boys’ in it, she wondered what part of speech to call the word ‘two’; it surely didn’t say anything about what the boys were like. So she asked her teacher, who simply said ‘Call it an adjective’. The grammar she was being exercised in was very superficial, of course. Being intelligent in her difficulty, it was no wonder that she took to philosophy later on. Plato was about the first grammarian (unless the Sanskrit grammarian Panini was earlier than he, which I think he wasn’t quite; but anyway there is no connection between them). Plato in the Sophist invented the distinction between a noun and a verb or predicate—what you are talking about and what your sentence is saying about it: όνομα and ρημα. He had earlier come to think that a sentence, a λογος, is a complex, a complex of names, συμπλοκη όνοματων. There are exceptions to this like a shout of ‘Fire!’ But it was a big advance in philosophy to see the essential complexity of most λογος.
Now a really serious and comprehensive book of grammar would treat numerals in a separate chapter. Here the more important part would be the grammar which expresses the essence of what we call, say, the natural numbers. Frege’s remark that a numeral can’t have the property of yielding itself when multiplied by itself is a grammatical remark; the child’s observation that ‘two boys’ doesn’t profess to say what any boys are like was a grammatical observation. It was in this way that Wittgenstein spoke of grammar and it is clearly an extension of the grammar you learn—if you learn any these days—as school children.
Now why is the assumption that ‘to understand’ stands for a process a grammatical illusion? Why should it be thought to be a process? May it not at least be an event? ‘I suddenly understood’ we may say. And don’t we also say ‘I began to gradually understand what he was talking about’? That sounds like a process, just as the other sounds like an event. But consider the following: ‘What are you doing? What’s going on here?’ Answer: ‘I’m working out the square root of 1729’; and compare it with: ‘What’s going on here? What are you doing?’ Answer: ‘Understanding the rules of chess’. That won’t do: one would have to say ‘getting to understand’. That takes time.
Lewis Carroll makes Alice say: ‘You can’t believe the impossible’ and the Red Queen reply: ‘With practice you can. With practice I can believe six impossible things before breakfast every day.’ This is a grammatical joke, a deep one, because it is about the depth grammar of the verb ‘to believe’.
The novelist Charles Dickens makes a contribution to the depth grammar of the verb ‘to mean’. Harold Skimpole orders lamb chops from his butcher to whom he owes a lot of money. The butcher sighs and says ‘I wish I meant chops the way you mean pounds’. ‘Oh, but you can’t’, says Skimpole. ‘You can’t mean chops and not give me them, because you’ve got chops. But I can mean pounds and not pay you pounds because I haven’t got any pounds.’
Now what is the grammatical illusion in regarding numbers as objects? People call Frege a Platonist partly for this view, and it’s not unreasonable to do so. The word in Greek would be which Plato said the forms are; Aristotle stole the word for his philosophy and it became the Greek we translate mostly as ‘substance’. Cabbages and cats and men are substances in that philosophy; Plato didn’t think they were —really real beings. Those were forms and though we learn that the Platonists didn’t think there was a form of number it is certain that Socrates is represented as thinking there was a form of two, three, four, etc.
That we speak of numbers, not numerals, when we say e.g. ‘2 is the only even prime’ is perfectly clear: Frege made it so, or re-established its clarity in what I have quoted. Are there essences expressed by the grammar of numerals? Surely. It belongs to the grammar of the term ‘numeral’ that a numeral names, designates, but is not, a number. Or if you prefer: is a sign for a number. And what is the essence expressed by the grammar of ‘Three’? Well, what it is a sign for has the property of being odd, not even; and this property does not change. That there is a procedure called ‘taking two away from it’ which we teach children and which leaves one. That it is the first number of the natural numbers such that a group of that number can be called ‘several’. That it therefore can be called the number of a group of things whose number can be seen and counted.
Several of the features of the grammar of ‘Three’ are common to many numbers, and sometimes what I have said sounds as if Three were indeed an object,—e.g. when I spoke of ‘taking two away from it’; doesn’t the ‘it’ suggest an object? Yes, it does—as a matter of superficial grammar. But when we think of the procedure and what a remarkable thing we teach a child to do, which we call ‘taking x away from y’, does it not appear that the initial appearance of an ‘it’ that one has done something to, is a sort of grammatical hallucination? I think it does and that this appearance is no illusion.
If the grammar of certain terms expresses essences, the mastery of their grammar is an indication of a grasp of essences, and essences can never be grasped except by intellect.
Now: was Wittgenstein an ‘essentialist’? To the extent that I have described, yes. But he toyed with the idea of peoples—tribes— whose languages contained expressions of different concepts from ours—colour-shape concepts, for example, without concepts of colour or of shape. Would they be missing something? Well, he remarks that you don’t always have a word for something just because you can see it. They’d be missing a hunk of language, but that doesn’t mean they’d be colour blind. Are we missing something because we don’t have colour-shape concepts? In this sort of questioning there is a suggestion that essences depend on grammar. He did not say: Essences ‘are created by grammar’, only ‘are expressed by grammar’. What is implied for my question I have to leave unanswered.