Why does Milinda step into Nagasena's trap?

I don't know which version/translation you're using but let's go with The Questions of King Milinda translated by T. W. Rhys Davids since it's nearly the first that Google returned.

The way I read it is this (let me paraphrase):

• Milinda: Hello your Reverence, what is your name?

• Reply: I'm known as Nagasena and people call me that, but it's just a name, there's no permanent soul in the matter.

  Milinda has been, presumably, brought up to assume there is a soul in the matter.

• Milinda: Hey guys, this Nagasena claims there's no soul in the matter!

  Milinda then tries reductio ad absurdum. Milinda tries to imply that, because it's obviously wrong to assume that Nagasena is hair, therefore Nagasena must be something else i.e. a soul.

• Milinda: Well look, Nagasena: if there's no soul, who gives gives you stuff, who enjoys the stuff you're given? Who lives the holy life? Or who is sinful?

  Here Milinda might be implying that Nagasena is (or is asking Nagasena whether he is) a 'nihilist'. Note that Buddhism describes eternalism (permanent soul) and nihilism (because nobody exists there are no consequences and no reason to be moral), as being opposite extremes and as both being wrong view.

• Milinda: So what is Nagasena then. Are you telling me that Nagasena is the hair on your head?

• Nagasena: I didn't say that.

• Milinda: Or the nails, teeth, etc.? The Skandhas? The combination of Skandhas?

• Nagasena: No.

Nagasena uses the chariot as an example; the chariot is:

• Not one of its parts
• Not any of its parts
• Not all the parts in combination
• Not something other than its parts

Milinda says that it's because of its parts that it's called a chariot.

Now to answer your question.

Why King Milinda does not consider all together - pole, axle, wheels etc. - his chariot?

Because of the context. Are you saying that the chariot has a permanent soul? That the soul of the chariot is the sum of its parts?

Because the chariot was being used as an analogy for Nagasena: Nagasena's parts, Nagasena's name, whether or not Nagasena had a soul, and whether "Nagasena" is just a name.

Normally one defines a chariot a means for transportation

So you think it would help to define a chariot by its function or use? Is that (the chariot's being used as a means for transportation) the chariot's permanent soul?

I personally found that kind of thinking (that objects have an eternal soul) useless and implausible when I read something like it in some book about Platonic Idealism (though I was a teenager at the time so I may have misunderstood it).

And, what by analogy would be Nagasena's function and use? Is that a permanent function or use, and/or a good way to understand that Nagasena has a permanent soul, and what the soul is?

OP: Why King Milinda does not consider all together - pole, axle, wheels etc. - his chariot? Normally one defines a chariot a means for transportation consisting of pole, axle, wheels, etc. This would be the correct answer to Nagasena's question what the chariot is.

Let's say King Milinda has a chariot with 4 wheels, pole, axle etc.

So, you can say a chariot is composed of all these parts.

But what if I remove just one wheel?

Does King Milinda ask, "oh... where is my chariot?"

No. He can still see his chariot with one wheel missing.

Has the chariot ceased being a chariot because one wheel is missing? No.

Then how do you define what a chariot is?

Different parts of the chariot are not the chariot by themselves. The combination of different parts also do not make up a chariot.

So, a chariot is just a label for a concept. There is no single concrete indivisible object called chariot. It is conditioned and compounded.

Similarly, the "self" or "being" are labels for concepts that are conditioned and compounded. The "self" and "being" are not single concrete indivisible things.

Maha-nidana Sutta: The Great Causes Discourse might help one to understand that rupa depends on nama, like nama depends on rupa, and as soon nama craves after a touch (rupa), a phenomena arises, something that is not really real but object of quarrel, Mr Jo Wehler. Has Jo Wehler given a good and complete description of the chariot or can it be rebuked?

As soon as an object is delineating and not seen as it really is, with right discernment, a world comes into existence including its observer. Is there a chariot if there is no contact? Is there a formation if there is no perception (imagination, remembrance)?

At least Milinda did not step into trap but got a way to unleash of it.

In addition, since just got across that, by trans. MN22:

SN 5.10 Vajira Sutta: Sister Vajira

Just as when, with an assemblage of parts, there's the word, chariot, even so when aggregates are present, there's the convention of living being.

Explaining with reference to this in MN22:, shows what the Ven. tried to give a chance:

SN 5.10 indicates that one of the ways of overcoming clinging is to focus on how the concept of "being" arises, without assuming the truth of the concept.

(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)

If I remember correctly, the traditional interpretation of this is that a pile of components on the ground is not the chariot either.

I suppose in modern language we could say that "Chariot" is a designation for the function that depends on the right configuration AND on the external supporting factors like the road, the horses, the gravity, the driver, the passengers etc. Plus, if individual components of the chariot are replaced with functionally equivalent but non-identical ones - the new chariot is neither the same nor different. Furthermore, the wheels etc. are only designations for things that can perform function of a wheel etc. - not to those particular wheels.

So the functional nature of the chariot arises from a number of functional conditions coming together - for which "the chariot" is only a label.

A similar illustration can be found in Vajira Sutta:

So you believe there's "a being"?
-- this is a view of Mara.
In a pile of mere components (sankhara)
no any such "being"'s found.

Just as they say 'chariot'
of an assemblage of parts,
So they'll call "a being"
the mere heap of components (skandha).

The key word here is "an assemblage" or in another translation "when the parts are rightly set". So that's why they are saying that the chariot is not all its parts together either - because the parts by themselves do not produce a function (of a chariot, a being etc), the right combination (of conditions) does.

But on the other hand, there is no "secret sauce" either - nothing apart from the factors and components that would set things in motion:

(from Viticcha Jataka)

The Ascetic tried to catch him in his words: "What is the Ganges? Is the sand the Ganges? Is the water the Ganges? Is the hither bank the Ganges? Is the further bank the Ganges?"

But the future Buddha said to him: "If you take exception to the water, the sand, the hither bank, the further bank, where can you find any Ganges River?"

The "neither found in any individual part, NOR in the pile of parts, NOR apart from the parts" thesis is probably constructed after canonical Buddha's line in which he shows how such phenomenon as the "Tathagata" can neither be found in the five skandhas ("piles") nor apart from them:

(From SA 105)

[The Buddha] asked again: "How is it, Seniya, is bodily form the Tathāgata?"

[Seniya] replied: "No, Blessed One."
[The Buddha asked again]: "Is feeling … perception … formations … consciousness the Tathāgata?"
[Seniya] replied: "No, Blessed One."
[The Buddha] asked again: "Seniya, is the Tathāgata distinct from bodily form? [32b] Is the Tathāgata distinct from feeling … perception … formations … consciousness?"
[Seniya] replied: "No, Blessed One."
[The Buddha] asked again: "Seniya, is the Tathāgata in bodily form?
Is the Tathāgata in feeling … perception … formations … consciousness?"

[Seniya] replied: "No, Blessed One."
[The Buddha] asked again: "Seniya, is bodily form in the Tathāgata?
Is feeling … perception … formations … consciousness in the Tathāgata?"

[Seniya] replied: "No, Blessed One."
[The Buddha] asked again: "Seniya, is the Tathāgata without bodily form … feeling … perception … formations … consciousness?"
[Seniya] replied: "No, Blessed One."

In my understanding, all of this is a finger trying to point out the fact that reality underlying conventional expression can never be exhaustively described, that our attempts to represent the reality is always an exercise in isolating individual factors and reassembling them into a model, like projecting a multidimensional figure on a two-dimensional surface - with a label (nama) stamped on top.

From taking our models seriously ("reifying" them) comes trying to make them our base, leaning on them, identifying with them, attaching to them - but because our models are merely designations for the fluctuating soup of factors - to assume them to be stable is an inherently faulty assumption - and from this comes dukkha. When we see reality as reality and models as models, we can stop confusing the two, stop getting attached to models, stop trying to establish our base in a model - so from this right vision (of the ineffable reality and the modeling activity) comes baselessness and from this baselessness comes cessation of nutrition of old dukkha and non-production of new dukkha.