Use of correlations between predictors How is the correlation between two predictors in a data set to be interpreted if there are numerous predictors in the data set? When calculating this correlation in a naive way by just considering the values of the two predictors across all of the rows, one is not controlling for the variations in the other variables. So is this correlation useful at all given that it contains "noise"?
I have the same question for the correlation between the response variable and any of the predictors. Since we are not controlling for the effects of variations in other variables, does this correlation have any significance at all?
 A: 
So is this correlation useful at all given that it contains "noise"?

A matrix of correlations between all the dependent variables is useful in the first stages of a data analysis, as it will identify those with very high correlations that might pose a problem with subsequent analysis - ie multicollinearity.

I have the same question for the correlation between the response variable and any of the predictors. Since we are not controlling for the effects of variations in other variables, does this correlation have any significance at all?

It depends somewhat on whether the goal of your analysis is prediction or inference. If it is prediction, then including the response variable can also be useful as it helps to identify what could be possible predictors in a further analysis such as regression. On the other hand if the goal is inference (ie gaining a better understanding of the underlying data generation process) then I would advise extreme caution. In that case, it is important to have some understanding of the underlying data generation process in order to avoid pitfalls such as including a mediator in a regression.
Correlation only measures linear associations. It says nothing about causation, and it says nothing about possible nonlinear relationships - see Anscombe's Quartet for a good illustration of the limitations of correlation.
A: 
is this correlation useful at all given that it contains "noise"?

The correlation matrix is used to check if there is any relationship between variables. Then you can remove duplicated variables or highly correlated pairs to reduce the dimensionality and make your modeling faster. It is not useful to find the noisy variables

Applications of a correlation matrix: 

There are three broad reasons for computing a correlation matrix:
  
  
*
  
*To summarize a large amount of data where the goal is to see patterns. In our example above, the observable pattern is that all the variables highly correlate with each other.
  
*To input into other analyses. For example, people commonly use correlation matrices as inputs for exploratory factor analysis, confirmatory factor analysis, structural equation models, and linear regression when excluding missing values pairwise.
  
*As a diagnostic when checking other analyses. For example, with linear regression a high amount of correlations suggests that the linear regression’s estimates will be unreliable.
  

ref: What is a Correlation Matrix?

I have the same question for the correlation between the response variable and any of the predictors. Since we are not controlling for the effects of variations in other variables, does this correlation have any significance at all?

It's called Bivariate analysis. According to Wikipedia:

Bivariate analysis is one of the simplest forms of quantitative (statistical) analysis. It involves the analysis of two variables (often denoted as X, Y), for the purpose of determining the empirical relationship between them.
  Bivariate analysis can be helpful in testing simple hypotheses of association. Bivariate analysis can help determine to what extent it becomes easier to know and predict a value for one variable (possibly a dependent variable) if we know the value of the other variable (possibly the independent variable)

ref: Bivariate analysis
