> > Of course I've heard of them. But I don't consider mid 20th century "modern" here. Science and mathematics have changed drastically in the last century: philosophy hasn't yet fully digested these developments. However, philosopher/mathematicians like Hersh have made very significant advances in "humanist" mathematical philosophy.
But these very comments confirm what I have been arguing. "Science and mathematics have changed drastically in the last century". Presumably you mean "in the last 40 years or so"? Well who could argue with that? Indeed, new theorems have been proved, problems solved, phenomena discovered. And yet, the sentence is pure vapour - rhetorics for the naive. In fact, nothing has occurred in mathematics that would in any fundamental way change our view of the nature of mathematics since Goedel (died 1978). The same is true for physics: the last discovery that fundamentally affected our understanding of the nature of the subject took place in 1972: it was the first of many experiments that confirmed the violation of Bell's inequalities - something that in my opinion totally undermines the view of physics you have been arguing for. About a year ago I attended a conference where the latest research on the fundamental issues affected by this were discussed (such as the notion of "independent reality") so can say with confidence that in all essential respects the situation remains as it was in 1972. So what are these supposed advances in mathematics and physics that have not been absorbed by " "humanist" mathematical philosophy?
And what does "humanist" mean in this context? All the philosophers I mentioned have been either scientists or mathematicians. Michael Polanyi was a great chemist, who would have probably won the Nobel prize (it was later won by his son John) if he did not at one point decide that philosophy was more important than chemistry (he was right then). Polanyi was, of course, a leading exponent of the view that intellectual (or aesthetic) judgement is the driving force of scientific discovery. You also mention Hersh, presumably Rueben, a mathematician, writer of popular articles and an amateur philosopher. Well it is well known that Hersh was an admirer of Imre Lakatos, who is well known for denying the distinction between mathematics and empirical sciences. He is also known for his denial that Darwin's theory satisfies his criteria for being considered "scientific" (see http://en.wikipedia.org/wiki/Imre_Lakatos)
> Quine is an interesting fantasist, but I can't consider him any more reliable as a guide to reality than, say, J. R. R. Tolkien.
Well, as far as I am concerned, he has a far better claim to being such a guide than anyone involved in this exchange.
>> > > The original issue of this thread was education. A philosophy that cannot distinguish between reality and hallucination is useless here. But, if we can understand mathematics as a human social construct, we can connect it to human social activities like education.
The issue was mathematics education of non-mathematicians. You extended it beyond its proper limits to the nature of mathematics and mathematicians. I consider this a worthless diversion. As far as mathematics is concerned the old adage has always been true: "Those who can, do. Those who can't, teach. Those who can't teach, teach how to teach."