Non-axiomatic definition of well-being I first wanted to ask this on philosophy@SE but, on a second thought, decided this might be a better place.

If we defined a general policy's objective being "to maximize well being and minimize suffering", we can say that "suffering" in that sentence is the opposite of well-being. But how do you define "well-being" by using lower level indicators, none of which are regressivly axiomatic and in a way that's so abstract that it's difficult to blame for cultural or any other kind of bias?

I would imagine that, at this point, defining well-being should be a province of economics and not philosophy. Do economists concern themselves with the question of defining well-being at an individual or collective level, by using natural/rationalist terms so that no amount of regression of the origins of any preceding term's definition runs into a proposition that will be considered "axiomatic"?
When you say that a policy's objective is to "maximize well-being", presumably you mean "maximize collective well-being".  And presumably, by "collective" well-being, you mean some sort of aggregate or average of the individual well-beings of all members of society.  So your question breaks into two parts:


What is the right measure of individual well-being?
What is the right way to aggregate individual well-being into a measure of collective well-being?


Question #1 is really a philosophical question (although many of the more philosophically minded economists have thought about it).  Broadly speaking, there are three families of approaches to defining/measuring individual well-being:


Hedonic:   Well-being is an emotional state (e.g. happiness).  Different hedonic approaches differ as to whether well-being is a short-term or instantaneous phenomenon (e.g. pleasure, absence of pain) or a more long-term one (e.g. "overall life satisfaction").
Preference-satisfaction:  According to this approach, well-being is the satisfaction of the individual's preferences.  To the extent that people want to be happy, and to the extent that they are happy when they get what they want, this overlaps the hedonic approach.  But there are situations where they come apart.  Also, different preference-satisfaction approaches differ over whether we care about the individual's actual preferences, or her "ideal" preferences (i.e. those she would have if she was perfectly informed, perfectly rational, etc.).
Perfectionist (or "objective list") approaches.  These approaches propose to measure well-being in terms of some list of objective criteria, such as physical health, education, material consumption, participation in the community, etc.  Whereas the two previous approaches are "subjective" (it is is up to individual to determine her own level of well-being), this approach is "objective" (the individual's own opinion is only one part of the data, and perhaps is entirely irrelevant).   So in an objective list approach, an individual could have a high level of well-being even if she herself thinks she is miserable (and vice versa).


Turning to the second part of the question:  what is the right way to aggregate individual well-being into collective well-being?  This is also a philosophical question (indeed, it is the central question of the entire "welfarist" approach to moral philosophy).  But it has also been explored extensively by economists.  Economists refer to such an aggregator as a social welfare function or social welfare order, and there is a huge literature on the axiomatic characterization of various families of social welfare functions. 

For a good introduction to these topics, I suggest the following books:


Handbook of Well-being and Public Policy, edited by Matthew D. Adler and Marc Fleurbaey,  Oxford University Press, 2016.   
Handbook of Rational and Social Choice, edited by Paul Anand, Prasanta Pattanaik, and Clemens Puppe, Oxford University Press, 2009.
Handbook of Social Choice and Welfare, volume I, edited by  Kenneth J. Arrow,   Amartya K. Sen and  Kotaro Suzumura,  Elsevier, 2002.
Handbook of Social Choice and Welfare, volume II, edited by  Kenneth J. Arrow,   Amartya K. Sen and  Kotaro Suzumura,  Elsevier,
2011.


Each of these handbooks contains chapters written by different experts on different aspects of these questions.  The first two books are the most "philosophical".  The second two books are more focused on axiomatic analysis of social welfare functions.  The very first book (the Handbook of Well-being and Public Policy) is probably the best place to start.