Weak And Stronger Platonists Viewed Through Systemic Functional Linguistics

Penrose (2004: 14-5):

It should perhaps be mentioned that, from the point of view of mathematical logic, the Fermat assertion is actually a mathematical statement of a particularly simple kind, whose objectivity is especially apparent. Only a tiny minority of mathematicians would regard the truth of such assertions as being in any way ‘subjective’ — although there might be some subjectivity about the types of argument that would be regarded as being convincing. 

However, there are other kinds of mathematical assertion whose truth could plausibly be regarded as being a ‘matter of opinion’. Perhaps the best known of such assertions is the axiom of choice. … Most mathematicians would probably regard the axiom of choice as ‘obviously true’, while others may regard it as a somewhat questionable assertion which might even be false (and I am myself inclined, to some extent, towards this second viewpoint). Still others would take it as an assertion whose ‘truth’ is a mere matter of opinion or, rather, as something which can be taken one way or the other, depending upon which system of axioms and rules of procedure (a ‘formal system’) one chooses to adhere to.

Mathematicians who support this final viewpoint (but who accept the objectivity of the truth of particularly clear-cut mathematical statements, like the Fermat assertion discussed above) would be relatively weak Platonists. Those who adhere to objectivity with regard to the truth of the axiom of choice would be stronger Platonists.

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From the perspective of Systemic Functional Linguistic Theory, on this view, the objectivity or subjectivity of mathematical statements is a matter of intersubjective (interpersonal) assessment. 'Stronger Platonists' are those who assess mathematical statements as unconditionally 'objective', whereas 'weak Platonists' are those who assess mathematical statements as 'objective', but with validity dependent on the starting assumptions and formal systems that are assessed as criterial for assessing validity.