I don’t particularly understand the following things:

1) what is meant by “we can only know what is required in order to secure the correspondence.” (green high light). Also, when I say “x can only” that means there are boundaries. What exactly are these boundaries, fleshed out?

2) “we can know the properties of the relations required to preserve the correspondence with sense-data” — what does properties of relations mean?

3) what point is being made about time-order seeming to have and really having the same order. He seems to say that orders stay the same. But for example, lightning and thunder, really happen simultaneously, but we perceive one after the other. Isn’t that a change in order?

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1.

we can only know what is required in order to secure the correspondence.

means we can only know what we perceive, we can only see an object the way we perceive it, we cannot necessarily know nothing what it is like in itself.

2.

we can know the properties of the relations required to preserve the correspondence with sense-data

Meaning here we can know what some things we "see" or "feel" through our sense-data, but we cannot grasp what the true form of them are, and cannot fathom to explain something as such to a person without such ability, such as a blind man born without vision, we cannot explain to him what something looks like for he cannot fathom it.

3.

what point is being made about time-order seeming to have and really having the same order.

The meaning behind "time-order" here is that while things may happen simultaneously, for example the lightning and thunder example here, is to say that there can be a difference between our sense-data of events and the physical science data behind them, When a lightning causes disturbance in the air around it, it instantaneously created a sounds, but when you are further away you experience the aforementioned sounds later, which is to say that "the order of events has changed" in your own sense data, such is indeed true, but in physical science data, it has not because it always starts from the same thing but we simply perceive it later depending on where we are.

Re. question 1, the preceding assumption is relevant since it makes clear the question is about the relation of map to territory, (private space to physical space).

"Assuming that there is physical space, and that it does thus correspond to private spaces, what can we know about it? We can know only what is required in order to secure the correspondence."

So I will complete the answer with a quote from Gregory Bateson:

What gets onto the map, in fact, is difference, be it a difference in altitude, a difference in vegetation, a difference in population structure, difference in surface, or whatever. Differences are the things that get onto a map.

A difference is a very peculiar and obscure concept. It is certainly not a thing or an event. This piece of paper is different than the wood of this lectern. There are many differences between them - of colour, texture, shape, etc... Of this infinitude, we select a very limited number which become information. In fact, what we mean by information - the elementary unit of information - is a difference which makes a difference. - (Steps to an Ecology of Mind, pp.457-459).

source: web archive

The differences are what is required to secure the correspondence.

Re. question 2: "What does 'properties of relations' mean?" Properties like altitude, a difference in vegetation, etc.

Re. question 3: "Lightning and thunder, really happen simultaneously, but we perceive one after the other. Isn’t that a change in order?"

Presumably Russell is saying that in the private space lightning and thunder is known to happen simultaneously, as they actually do in physical space.

Hence "the time-order which events seem to have is, so far as we can see, the same as the time-order which they do have."

He also says "it must not be supposed that the various [phenomena] have the same time-order as the sense-data which constitute the perceptions of those [phenomena]."

So you hear lightning and thunder separately, you are aware they happen simultaneously, and in physical space they do happen simultaneously.

The key-concept is acquaintance :

We shall say that we have acquaintance with anything of which we are directly aware, without the intermediary of any process of inference or any knowledge of truths. (Russell , The Problems of Philosophy (1912)).

Physical distance, like any other scientific concepts, is not an immediate sense-data.

We do not perceive distance; we infer it through a process: measuring it.

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