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I do not understand the basis of one of Russell's claims at the end of the chapter 'Similarity of Relations' in his Introduction to Mathematical Philosophy. I have taken an excerpt and emboldened the parts that I do not follow.

[...] it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, [...]. Where such hypotheses are made, it is generally supposed that we can know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. [...] In short, every proposition having a communicable significance must be true of both worlds or of neither.
(page 61)

I don't understand how Russell can say that the objective world will necessarily have the same 'structure' as the subjective world.

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    It's worth noting that when Russell states "if the hypothesis as stated were correct" he is NOT talking about his own hypothesis.
    – MmmHmm
    Mar 16, 2017 at 13:45
  • The proposition that follows "if" is a hypothesis and asserts nothing - this is a notion implicit but fundamental in all of Russell's writings
    – George Chen
    Mar 16, 2017 at 14:20

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the objective counterparts would form a world having the same structure as the phenomenal world

This is just saying that if you make a true statement "about the world" it is necessarily given more or less directly in terms of our perceptions.Since the statement is true there must be something "out there" (responsible for our perceptions) that has the kind of structure given in the statement. So we have the structure indicated by our statement and some external arrangement of 'stuff' acting in the manner described by the statement.

In short, every proposition having a communicable significance must be true of both worlds or of neither.

So whatever we say in a statement must apply to the world of experience and the objective world and if true in the world of experience is also true for whatever 'stuff' it is describing.

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  • Ok I follow up to "out there". My question is, how do you know that the thing "out there" has the same structure as what we perceive?
    – lagrange103
    Jun 24, 2016 at 0:35
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    The presumption here is that there is a "true statement" about the world. Some description of our experiences is taken to accurately describe what we experience. Given that the description accurately describes some relationships between our perceptions then something "out there" corresponding to those experiences must be causing them and the relationship as described.
    – Vector Shift
    Jun 25, 2016 at 1:19
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    This is rather tautological. All we can ever know about the world is by experiences so a true statement of relationships based on perceptions actually defines the relations of the objective world. If you want some way to discuss relationships in the objective world without basing that on experiences then it's hopeless. That is the failing of metaphysical distinctions which can't be settled by observations/experiments.
    – Vector Shift
    Jun 25, 2016 at 1:23
  • I think Russel is following a empirical view in this regard.
    – Mockingbird
    Mar 28, 2017 at 12:01
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He says:

it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding with the differences in the phenomena to which they give rise.

He assume that the "phenomenal" world is the counterpart of a "noumenal" world that "causes" it and that the two share the same structure.

Consider the previous discussion about "structure":

We may say, of two similar relations, that they have the same “structure.” For mathematical purposes the only thing of importance about a relation is the cases in which it holds, not its intrinsic nature.

Thus, if all that matters is the structure and the two worlds share it, the idea of a "noumenal" world completly unknown to us if not by way of his "similar" phenomenal manifestation loose much of its interest.

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  • I cannot understand people who 'ontologise' the noumenal. I think most of the reception of Kant's philosophy in anglo-saxon philosophy is or at least has been deceived by Russel's misreadings.
    – Philip Klöcking ♦
    Jun 21, 2016 at 10:54
  • @PhilipKlöcking - maybe... but there is a strong opposition in modern anglo-saxon philosophy against Kant and "idealism" due to the common sense realism; see Moore and the refutation of Idealism: "That ‘to be true’ means to be thought in a certain way is, therefore, certainly false. " And see Moore on Common Sense and Certainty.
    – Mauro ALLEGRANZA
    Jun 21, 2016 at 11:31
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    Does Russell give any other explanation or evidence to the claim of isomorphism? The way I see it, he is asserting a bijection between the objective and the phenomenal. How does he know it's not a surjective relationship, for example?
    – lagrange103
    Jun 22, 2016 at 1:05
  • @PhilipKlöcking whatever do you mean by "ontologise"? As in "reasoned argumentation regarding the ontological status of the noumenal"? As is, Kant does a great job already of making some glaring errors. You should read Searle's summary of Kant's Critique of Pure Reason.
    – MmmHmm
    Mar 16, 2017 at 13:29
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    @MauroALLEGRANZA Russel doesn't state that he presumes this, he states that if this is presumed, then such is the logical conclusion. "if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world"
    – MmmHmm
    Mar 16, 2017 at 13:32
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Strictly speaking, the hypothesis is this: Objective things are mapped into phenomenal things by the same correlator S (see beginning of this chapter). If this hypothesis is true, then the phenomenal world share the same structure as the objective world, and we can thus penetrate phenomena and know some truths in the objective world.

This hypothesis is not always true. Take thunder and lightening for example. They are caused by the same explosion in the objective world but are separate events in the phenomenal world because the correlators are different: one maps the explosion into a visual sensation; the other,into an audio sensation.

The ground breaking point Russell tries to make through is that * it is possible to know some truths in the world of ding an sich.*

Things in themselves used to be thought unknowable. "This apple is red" is not about the apple in itself because colour exists only in the mind and red is not an intrinsic property of the apple, thus all we can say about the apple are the sensations caused by the apple.

But When you see an earwig inside your living room, you know for sure that the relation, inside, which the earwig has to the living room is an intrinsic property of things in themselves - this relation belongs to the class of relations Russell called structure, and Russell demonstrated that structure in the objective world are actually knowable to us.

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