Bayesianism involves taking Bayesian statistics seriously 
really seriously. Bayesians typically hold that:
 Probability should always be used to represent uncertainty;
 There's no uncertainty, except inside some kind of mind;
 There's no such thing as absolute certainty: beliefs are always uncertain;
 It is possible to make probability assessments without any evidence;
 Bayesian statistics is an important foundation of the scientific
method.
Bayes rule can be interpreted as indicating how to update
probability estimates in response to observational evidence.
This operation is an important part of the scientific method.
It doesn't tell you which experiments to perform or how to
get initial probability estimates (or "priors", as Bayesians
call them), but it does specify how to update your world view
in response to observations.
The idea that probability represents uncertainty goes
counter to most formal education in probability theory  which
holds instead that probabilities represent frequencies
of occurance: the "frequentist" interpretation of
probability. For frequentists, probabilities are
observer independent. A coin flip might have p(heads) = 1/2 regardless of the observer. Prior probabilities don't matter.
The observer's ignorance doesn't matter. p(heads) = 1/2 is an observerindependent fact.
This interpretation of the term "probability"
is an irritation to Bayesians. There should not be two
meanings of the term "probability". Bayesianism is much
more important and significant than frequentism.
It should get monopoly rights over the term "probability".
Historically, prominent opponents of Bayesianism have included
Karl Popper and his followers. Popper denied the validity
of induction, and held that only falsification was
significant. By contrast, Bayesian statistics uses the
same mathematical approach to represent both falsification
and confirmation.
Observations that falsify a hypothesis are typically
stronger evidence than observations that confirm it 
but both types of observation result in probability
updates via the same formula: Bayes theorem.
Another casualty of Bayesianism is David Deutsch's
"Constructor Theory". This holds that we can characterize
physical laws by identifying which transformations are
possible and which transformations are impossible.
However Bayesianism insists that we can never be certain about such
things. A supposedly impossible transformation might one day turn out
to be possible. Similarly a transformation we believed to be possible
might turn out to not be reproducible. The initial belief in its
possibility could have been due to experimental errors.
Bayesianism insists that statements about "possibility" and
"impossibility" be reformulated as assessments of probability 
turning the world into shades of grey.
It is possible to rework "Constructor Theory" in Bayesian terms
by considering instead the probabilities associated with
various different transformations. However, it then sounds
much more normal and much less revolutionary.
David Deutsch himself frames his ideas as conflicting with Bayesianism. He writes:
A scientific explanation is a statement of what is there
in reality, and how it behaves and how that accounts for
the explicanda. Neither confirmation nor credence nor
‘inductive reasoning’ (from observations to theories or
to justifications of theories as true or probable)
appear in this account. [...] This contradicts the
‘Bayesian’ philosophy that rational credences obey
the probability calculus and that science is a
process of finding theories with high rational credences,
given the observations.
It's generally desirable to quantify uncertainty 
as Bayesianism recommends. Without quantification,
it isn't possible to update properly on incoming evidence.
References
