Inductive Probability in Mathematical Theory In M.G. Bullmer's Principles of Statistics (Dover edition 1979, though seems like nothing significant was changed from the 1967 second edition), he states in the conclusion to the first chapter that:

"It has been reluctantly concluded by most statisticians that inductive probability cannot in general be measured and, therefore, cannot be used in the mathematical theory of statistics...it does not seem possible to construct a numerical scale of such (inductive) probabilities."

This sentence struck me as odd, since Wikipedia's article on Inductive Probability seems to have a wealth of information referencing Bayes Theorem, Inference, and applications in A.I/Machine Learning.
Is this due to the fact that the field of Inductive Probability has grown significantly since the 1960s/1980s?  Since I'm learning probability mainly for applying it to machine learning should I pick up another book on statistics, or are the fundamentals still helpful when applied to ml?
Or is Bullmer's statement still correct? I guess I'm not really sure what a "mathematical model" is and whether the information in the wikipedia article qualifies as such.
 A: The Wiki article referred to says:
"Inductive probability attempts to give the probability of future events based on past events. [...] Bayesian inference broadened the application of probability to many situations where a population was not well defined. But Bayes' theorem always depended on prior probabilities, to generate new probabilities. It was unclear where these prior probabilities should come from."
So, even the Wiki article is reluctant about the source of information when telling about prior probabilities.
Here is an example to enlighten the problem with unknown priors:
Assume that we have an unfair coin which we intend to flip twice independently. Let the probability of the heads denoted by $p$ -- this is our prior information. What is the probability that we get heads twice?
The answer seems to be easy: $p^2$ ; what else?
Now, what if we do not know $p$? Then we may assume the existence of a random variable $P$ uniformly distributed over $[0,1]$. So, we imagine that first we (will) observe the event $P=p$ (for some $p$) then we will use this $p$ during  our calculations. In order to do any calculation we need another innocent  looking assumption: Let the two coin flips conditionally independent given the event that $P=p$.
Now,
$$P(head_1 \cap head_2\mid P=p)=p^2,$$
$$P(tail_1\cap tail_2\mid P=p)=(1-p)^2$$
$$P(tail_1\cap head_2\mid P=p)=P(head_1\cap tail_2\mid P=p)=p(1-p).$$
(I hope that my notation is not confusing so far.)
The following calculations will be as easy as confusing:
$$P(head_1)=P(head_2)=\int_0^1 P(head_1\mid P=p) dp=\int_0^1 p \ dp=\frac 12,$$ 
$$P(tail_1)=P(tail_2)=\int_0^1 P(tail_1\mid P=p) dp=\int_0^1 1-p \ dp=\frac 12,$$
$$P(head_1\cap tail_2)=\int_0^1P(head_1\cap tail_2 \mid P=p) dp=$$
$$=\int_0^1 p(1-p) dp=\frac 16\not =P(head_1)P(tail_2).$$
That is, our coin flips turned out to be not independent...
Our assumption about the distribution of the random variable $P$ and the assumption of conditional independence of the coin flips lead us to a contradiction with our most natural hypothesis that the coin flips are independent. This is just a simple example to explain why statisticians are getting reluctant when using traditional probabilistic reasoning in certain cases. 
