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Wittgenstein on symbols

Why does Wittgenstein have a problem with logical statements saying nothing ? (5.5303) . How would Wittgenstein want us to interpret f(a,a) ?

He also mentions axiom of infinity from which Russell argued for the existence of an infinite number from the definition of cardinal number. He defined 0 as number of propositions that are true and false or the number of elements in an empty set. Then the set containing only 0 has 1 element. Hence the number 1 exists and other natural numbers can be easily proven to exist. Does avoiding the use of equality symbol prevent us from equating objects that are not identical or especially in case of Aleph numbers, say equating the set of odd numbers with the set of even numbers since they have the same cardinality. For example in case of 1=p and 1=q , we conclude p=q but p and q are not identical. Which may cause problems according to wittgenstein as he writes

" Roughly speaking: to say of two things that they are identical is nonsense… "

I am really confused here and l am quite sure that l am getting this wrong completely or missing the key issue beforehand. It would be great to see some clarification regarding the points shown in the picture. I am quite happy to change my mind if l see some interpretations that are polar opposite of what l have understood.

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  • It seems to me that if a = b then they are different and thus not equal. Usually we ignore their differences in order to focus on their sameness, but taken as whole 'a' cannot equal 'b'. It can only equal a. The issue is the identity of indiscernables. . – PeterJ Jan 20 at 12:43
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Because he reads a and b occurring in the atomic proposition f(a,b) (e.g. "a is to the left of b") as referring to two different objects.

According to Wittgenstein, the only legitimate use of the sign of identity is at a meta-level, in order to talk about the use of signs, and not to assert anything substantive about the world. Thus he says:

4.241 When I use two signs with one and the same meaning, I express this by putting the sign ‘=’. between them. So ‘a = b’ means that the sign ‘a’ can be substituted for the sign ‘b’. (... like Russell, I write the equation - definition - in the form ‘a = b Def.’ A definition is a rule dealing with signs.)

4.242 Expressions of the form ‘a = b’ are, therefore, mere representational devices. They state nothing about the meaning of the signs ‘a’ and ‘b’.

If so, at the "object language" level, statements with identity are non-sense: a=a is uninformative (it is trivially true... but it is not a tautology, and this is a problem for Wittgenstein view!) and a=b is simply false: different names must mean different objects.


The "problem" with the axiom of Infinity regards W&R's project, developed into the Principia, to ground mathematics on "pure" logic. But the assumptions that there are an infinity of objects in the universe is not something "purely" logical.

The "solution" Wittgenstein hints to amounts to saying: we have a language with an infinity of individual constants that refer to different objects...

But in this way we assume again something about the world that can be true or not, but surely that cannot be asserted on the basis of logic alone.

  • Could you elaborate on how this proposal tackles the problem that appear from axiom of infinity. There must be some objections to writing f(a,a) instead of f(a,b).a=b . – Aristotle Stagiritis Jan 19 at 15:16
  • Interesting. Makes sense in a way. Wittgenstein always had a keen sense of making statements that transcend the discussion beforehand and are also very difficult to refute. – Aristotle Stagiritis Jan 19 at 16:01
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It helps to put this in a concrete example. If we have two apples sitting on the table before us, it is self-evidently true that they are not the same apple, so saying apple A equals apple B (a=b) is nonsense. If we have one apple sitting on the table before us, then it would be pointless to say that apple A equals apple A (a=a); that is a best a trivial tautology and at worst a circular definition.

I think Wittgenstein is referencing Frege's morning-star/evening-star paradox (See Russell's On Denoting). The problem is that we have two terms that supposedly denote two different objects — the morning star and the evening star — then we discover that (in fact) they both point to the planet Venus. That posed a problem for analytical philosophy, which depended heavily on strict denotation (a strong connection between propositional terms and real-world referents). Saying A=A or B=B is qualitatively different than saying A=B, because the first is tautologically true while the last requires a detour though extended logic, observation, or etc. Frege wanted to say that concepts had both denotation and sense: denotation being the unique thing itself; sense being the multiple ways it can present itself to the subjective mind. Russell objected to the idea of introducing a subjective construct, and tried to argue for A=B on the grounds of indirect denotation, which was not particularly convincing. Wittgenstein is saying that the paradox arises in the first place because we've imposed this weirdly abstract concept of 'equals' where it doesn't belong.

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    If l am not mistaken. The stance of Wittgenstein would not allow us to give definition of numbers in the following manner, " 0 is the number of elements in an empty set" Since we would need to use the = symbol. I think the only possible way left would be to take 0 as a given and then progressively build more natural numbers. – Aristotle Stagiritis Jan 19 at 17:49
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    @AristotleStagiritis — I don't think that's quite right. W wouldn't disagree with mathematics and set theory as an abstract system (a game, in his later language, with its own set of rules). He was more concerned here with logical propositions and identities/referents. I think he'd be fine with "0 is the number of elements in an empty set" so long as we acknowledged that the token 'an empty set' has no real-world referent. – Ted Wrigley Jan 19 at 18:05
  • I still have a few doubts with regard to how far Wittgenstein thinks the symbol is meaningless. Logical propositions such as a=a are self referential and do not refer to anything out there in the world yet he still seems to have a problem with that. Yet l understand his first objection of calling two things identical as nonsense. Btw Thanks for the feedback – Aristotle Stagiritis Jan 19 at 18:56
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    @AristotleStagiritis It is one thing to say "0 is the number of elements in the empty set" and claiming that any one zero is truly identical with every other instance of zero. Things can be equivalent with respect to a given role, e.g. countiing, without being exactly identical. The empty set of round triangles and the empty set of purple ideas are equally large, but they have different potential referents. In a programming language sense, they are empty collections of incompatible types, so they are not equal. – hide_in_plain_sight Jan 20 at 3:59
  • (Sorry, "...of an empty set". The idea that there is "the" empty set is exactly where we would come into trouble.) – hide_in_plain_sight Jan 20 at 4:14
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It is proper to say that a and b satisfy a the equivalence relation that defines a field of inquiry, but equivalence is all there is, not equality.

Two infinite sets are not equal on account of having a bjiection, instead, they satisfy the equivalence relation of cardinal counting. The odd integers simply are not the even integers. 1 + 1 does not equal 2, the cardinalities combine in the expected way. But cardinality is not a property, or a function, it is an equivalence relation. (Properties have domains, functions have images, cardinality has neither.)

Clearly setting the collection of groups and the collection of ordinal numbers side-by-side gives you two things. But you cannot put those things into an object that represents the pair and compare it to the set {0, 1}. The objects involved are too slippery. If you let them be contained, then you can pull each one of them out as a set, and you would have the Burali-Forti paradox. (Nor can you properly capture the notion of cardinality as a set of ordered pairs, because of examples like this, so it calls for a more general notion of relation than a set of ordered pairs.)

Wittgenstein's approach is the first that identifies the real importance of equivalence relations, over the notion of reifying collections and declaring the collections associated to different objects identical.

The former is real, the latter is cheating. It is a shorthand that degrades meaning.

To pull it forward into more modern terms, for instance, we cannot say, in a ZFC sense, that 1 is the collection of all singletons, 2 the set of all pairs, etc. Some singletons are not containable. The collection of ordinals is singular, but it cannot be a member of a set. Nor can we say 1 really is {0}, only that {0} is our chosen representative of the cardinality of 1.

Not using equivalence explicitly, is not really consonant with the idea that not all collections are sets -- which everyone has to accept in order to avoid paradox.

The notion of equality is artificial, and needs to be avoided if your domain of application is not carefully circumscribed by a given equivalence dictated by the context (i.e. Real numbers as convergent sequences of rationals.) Logic as a whole is not circumscribed as a field in this way. There is no domain that classifies all the things we want to talk about.

  • To define natural numbers, the convention is to take 0= { } then 1={ { },0 } and so on. So this notion is in use in modern mathematics. Would Wittgenstein have problems with this. Albeit it is used in definitions . ( I think he allows it for definitions and not for proposition. ) The link is below math.stackexchange.com/questions/123831/… – Aristotle Stagiritis Jan 20 at 4:11
  • That is one convention, but it is kind of wrong to think that is how they are naturally defined. He would not have a problem with this notion as mathematics, but as logic it is not proper. That is just not what everyone actually means by 1. To use this as a definition in ordinary life, is injecting a sophisticated, overpowered move from an alien expert domain into a more basic game that is more broadly shared. – hide_in_plain_sight Jan 20 at 4:20
  • So, yes, he would object, and the older Wittgenstein actually figured out why he would object... He introduces the notion of reaching across the boundaries around interlocking language games when he discusses, for example, "God talk." And this is kind of an instance of such a problematic action. This is not a definition anyone would accept and could use when one first needs to know what numbers are -- so it cannot really be the definition in a proper sense, even if it is the best way to formulate the definition when talking about metamathematical theories in the most concise way possible. – hide_in_plain_sight Jan 20 at 4:23
  • In math, two definitions that meet a given relation are the same. But that is an equivalence and not an equality. The everyday definition and this abstract construction are equivalent in mathematics but they are clearly different in reality – hide_in_plain_sight Jan 20 at 4:29
  • @MauroALLEGRANZA Evidence? It is what mathematicians often take away from this argument. – hide_in_plain_sight Jan 20 at 12:08

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