Penrose’s arguments for Platonism, with commentary:
“Our individual minds are notoriously imprecise, unreliable and inconsistent in their judgements”
“Road to Reality”, Roger Penrose, p12-13
So when we do maths we are not using our minds, but something else? Or maybe we are just using our minds in disciplined way—after all, the discipline of maths has to be painfully learnt.
“Does this not point to something outside ourselves with a reality that lies beyond what each individual can achieve?”
What is achieved by individuals is achieved by individuals. How can some transcendent realm speak through the mouth of an individual?
“Nevertheless, one might take the alternative view that the mathematical world has no independent existence, and consists of certain ideas which have been distilled from our various minds, and which have found to be totally trustworthy, and are ‘agreed by all’, for example, or ‘agreed by those in their right mind’s, or ‘agreed by those who have a PhD in mathematics’ (not much use in Plato’s day), and who have a right to venture an authoratiative opinion”.
Which mathematical world? The world of mathematics so far proven and understood, or the world that is waiting to be explored? However few mathematicians there are, there is no contradiction that they carry the world of known mathematics around in their heads. The fewer there are, the less there is to carry round. Perhaps Penrose thinks they must carry the whole Platonic world around to act as a “standard”, but mathematical proof doesn’t work by direct inspection of Platonia, it works slowly and painfully by axioms and deductions.
“There seems to be a danger of circularity here; for to judge whether someone is in his or her right mind requires some external standard”
External to them, yes...but when we make such judgements, we ue finite and earthly resources. The idea that Platonia provides a once-and-for-all absolute standard is not of much use unless it can be explained how the standard is applied. The actual standard against which a proposed arguemnt is tested involves the community of mathematicians checking it with their flawed and finite minds. Platonia might provide a higher standard, but it is not one anyone has ever succeeded in employing.
“What I mean by this existence is really just the objectivity of mathematical truth”.
And what does thatmean? Existence is existence, truth is truth. People who disbelieve in Platonism can still believe in the epistemic objectivity of mathematics.
Penrose’s arguments for Platonism, with commentary:
“Our individual minds are notoriously imprecise, unreliable and inconsistent in their judgements” “Road to Reality”, Roger Penrose, p12-13
So when we do maths we are not using our minds, but something else? Or maybe we are just using our minds in disciplined way—after all, the discipline of maths has to be painfully learnt.
“Does this not point to something outside ourselves with a reality that lies beyond what each individual can achieve?”
What is achieved by individuals is achieved by individuals. How can some transcendent realm speak through the mouth of an individual?
“Nevertheless, one might take the alternative view that the mathematical world has no independent existence, and consists of certain ideas which have been distilled from our various minds, and which have found to be totally trustworthy, and are ‘agreed by all’, for example, or ‘agreed by those in their right mind’s, or ‘agreed by those who have a PhD in mathematics’ (not much use in Plato’s day), and who have a right to venture an authoratiative opinion”.
Which mathematical world? The world of mathematics so far proven and understood, or the world that is waiting to be explored? However few mathematicians there are, there is no contradiction that they carry the world of known mathematics around in their heads. The fewer there are, the less there is to carry round. Perhaps Penrose thinks they must carry the whole Platonic world around to act as a “standard”, but mathematical proof doesn’t work by direct inspection of Platonia, it works slowly and painfully by axioms and deductions.
“There seems to be a danger of circularity here; for to judge whether someone is in his or her right mind requires some external standard”
External to them, yes...but when we make such judgements, we ue finite and earthly resources. The idea that Platonia provides a once-and-for-all absolute standard is not of much use unless it can be explained how the standard is applied. The actual standard against which a proposed arguemnt is tested involves the community of mathematicians checking it with their flawed and finite minds. Platonia might provide a higher standard, but it is not one anyone has ever succeeded in employing.
“What I mean by this existence is really just the objectivity of mathematical truth”.
And what does thatmean? Existence is existence, truth is truth. People who disbelieve in Platonism can still believe in the epistemic objectivity of mathematics.