# Can anyone help me understand what does Wittgenstein mean by "multiplicity" (logical/mathematical) ?

I would not presume to say what Wittgenstein meant by "multiplicity", but am glad to share what it brought to my mind:
1. In trying to find a synonym for a word, the thesaurus has a range of meanings, some of which are not near my intent, but carry one of the uses of that idea; the legitimacy of all the synonyms is multiplicity to me.
2. A lexical definition of a word often needs the original meaning (etymology) to get full appreciation of why that word exists.
3. In diagnosing a patient, a physician needs to evaluate the symptoms from the patient's personal experience, and evaluate it in terms of the physiology of the affected system, and evaluate it in the context of the chemicals in that system, and evaluate it in terms of how it operates psychologically and socially.Holding all those factors in mind is applying multiplicity.
I hope this helps.
Multiplicity has the following meanings:
1. Many.
2. arithmetic multiplicity [or in mathematic multiplicity]
3. Isomorphism between two complex structures A and B [Wittgenstein].
4 ...

In the following examples, I will illustrate the third meaning that Witt often uses.
Example 1: You have
A consists of A1 = {1, 2, 3}, A2 = {4, 5}, A3 = {6, 7, 8, 9}
and
B includes B1 = {a, b, c}, B2 = {d, e}, B3 = {f, g, h, i, j}

When there is a one-to-one corresondence between A and B:
A1 <-> B1
A2 <-> B2
A3 <-> B3
and
1 <-> a
2 <-> b
...
9 <-> j
then we say A and B have the same 'multiplicity'
Example 2: Assume we have commands
A means +
B means =
so
ABABBA
has the same 'multiplicity'with
+ = + == +

and
ABABBA has not the same 'multiplicity' with
+ = + = + =
In Mathematics, multiplicity is the number of times a particular member of a set appears in that set. For instance if a specific solution to a polynomial function occurs twice, it has a multiplicity of 2.
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