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Leslie Smith wrote: - the computer-proof the 4-colour problem: is it mathematically valid? If it is, it is a necessity. It the proof is invalid, it's not. Whatever feelings I or you may have about this proof are irrelevant as to whether the proof really is valid or invalid. Les and anyone else interested, Yes, we do disagree.?I think the above leaves out phenomenological aspects that we do well to include in our theorizing.?In that sense I think it is too objectivist. How do we know whether a mathematical proof is valid or not??Say that we consider it a matter of meeting certain mathematical/logical?criteria.?What is the basis of our acceptance of the criteria??I claim that our acceptance involves an experience we have of logical power.?I use the term "experience" in the phenomological sense.?I hold that without an internal experience of logical power, our acceptance of criteria of mathematical/logical validity would be either totally outside our consciousness or conventional. Furthermore, I hold that our experience of logical power varies according to what frames we have constructed.?Using frames of level X one is not able to follow certain arguments even though one can understand each statement on its own--because to experience the logical relations among the statements, one needs frames of a higher level.?Here "experiencing the logical relations" is largely a reformulation of more standard expressions such as "understanding the logical relations" or "appreciating the logical relations." Moreover, I hold that people often attribute their confidence in some conceptual activity to its empirical success, and do not recognize the role of relevant internal experience of logical power.?For example, I claim that many people consider that they have confidence in the procedure of counting because they have lots of empirical evidence that it works fine.?In contrast, I claim that their confidence in counting crucially involves their internal experience of its logical validity. Also, I hold that in the phenomenological sense, we experience the environment in terms of objects, and not just individual sensations.?We do not simply in some more abstract way "know" that the are objects out there.? This is similar to hearing a melody and not just the individual notes--we experience the melody, not just more abstractly know that it is there.? Similar to both is our experience of the logical power of an argument and not just the individual statements.? Our experience of the environment in terms of objects and our experience of logical power are deeply related.?I consider that whether or not Piaget explicitly went down this road, he opened the door for us to do so.?Yet, doing so involves more emphasis on the phenomonology than has been usual in constructivist thinking. I add that I consider internal experience of logical power to be open to influence by the cultural environment.?One can develop an expertise relative to this internal experience, as one can in areas more generally considered in phenomenological terms.? I hold that the development of such expertise involves a dialectic between the individual's construction of frames and cultural practices involving external representational systems.? A useful theory of consciousness will need to articulate this in a productive manner.