What will change everything is radical paradigm shift in the scientific method that opens up horizons beyond the reach of Boolean Logic, Digital Manipulations and Numerical Evaluations.
Due to my advanced age, I am not likely to witness the change. But I am seeing signs and have my hunches. These I will briefly spell out.
To change Everything a radical paradigm shift must interrupt the scientific method's race:
STOP for a moment's reflection; what are you up to?
How do you know your dog would rather be a cat — just because you prefer cats. Did you asked him, have you figured out how to ask him?
Having figured out how to do something is not enough reason for actually doing it. That's one aspect of the paradigm shift I am expecting; coming from inside the ranks. Evaluation of scientific results and their potential effects on the world as we know it is of particular urgency these days when news are spreading so easily all over the population. Of course we do not want to regress to a system of classified information that generates elitism. Well this problem is creating the, not so new anymore, philosophical discipline of applied Ethics; if only it keeps scientifically well informed and focused down to earth on concrete issues.
The goal of this part of the shift is a tightening of the structure of the whole conglomerate of the sciences and their presentation in the media.
But this brings me to the more radical effect of the shift I am envisaging; a healing effect on the rift between the endeavors that are bestowed the label "scientific" and the proliferation of so-called "alternative" enterprises, many of which are striving to achieve the blessings of scientific grounding by experimentations, theories and statistical evaluations, whether appropriate or not. A true and fruitful symbiosis that leads to a deeper understanding of the meaning of Human Existence than the models of a machine or of a token created by a Superior Being for the mysterious purpose of suffering through life in the service of His Glory.
Where do I expect the decisive push to come from? Possibly from the young discipline of cognitive science, provided psychology, philosophy and physiology are ready to cooperate. There are shoots rising up all over, but I won't embark on a list. Once the "real thing" is found or constructed it will be recognized. It will have shape and make sense.
The myth of the scientific method as the only approach to reality will become obsolete without loss to man's interaction with this world. The path to understanding has to be prepared by a direct, still somewhat mysterious approach of hunches and intuitions in addition to direct perceptions and sensations. Moreover the results of that procedure are useless unless suitably interpreted.
Well, this is as far as I am ready to go with this explanation of a hunch. The alternative to my current vagueness would be rigidity prone to misinterpretation.
As to my own turf, Mathematics, I do not believe there will be any radical change. Mathematics is a rock of a structure, here to stay. Mathematical insights do not change, they become clearer, dead ends are recognized as such, but what is proved beyond doubt what is cumulative.
But methodological changes here are in order as well as meta-mathematical and philosophical interpretation of the nature of results. So is the evolution of an ever more lucid language.
I personally believe we'd do well to focus on Mathematical Intuitionism as our Foundations. Boolean thinking has done its service by now.
To sum up what I am expecting of this paradigm shift are: clarification, simplification and unification of our understanding and with it the emergence of a more lucidly expressive language conducive to the End of Fragmentation of Knowledge.