[For health reasons my activity on this blog has been very low, sorry. I owe readers at least a number of follow-ups, but I await some health improvement before doing so.]
The below post which I contributed was rejected by the moderator of the FOM-forum. Immediately afterwards the thread was closed… 🙂 see closure message. This to me illustrates the social aspect of mathematics mentioned below. I put the post here, knowing it creates yet another thread which I should follow up…:
—message sent to FOM Thursday 6 June 2013— Re: Psychological basis of intuitionism
Perhaps the subject would be better phrased thus: `Psychological basis of (what is perceived as) scientific/mathematical truth´.
This issue has a long history, with much static in the communications, if I may put it like that.
It seems to me that these are some key elements:
* Mathematical platonism and Constructivism
* Science (and mathematics) as a mental activity
* Science (and mathematics) as a social activity (including group mechanisms and communication issues)
* What is mathematical reasoning? What is truth?
* What is the role of psychology and philosophy in science/mathematics?
The large majority of mathematicians are unaware, I believe, that they adhere to mathematical platonism. Mathematical platonism is what we are taught (mostly subconsciously) in high school already, and is continued in most academic mathematical traditions, without being explicitly mentioned.
It would perhaps be refreshing to rename Constructive mathematics as `non-Platonic mathematics´. In this way a comparison can be made to Euclidean and non-Euclidean geometry. You will recall that it literally took the mathematical community centuries to allow for just the possibility of non-Euclidean geometry (let alone its practical use). The psychological basis of that reluctance came, in my humble opinion, from the strong connection that mathematical platonists tend to make between idealized mathematics and `absolute/objective scientific truth’.
[[Personally, I do not believe in `objective truth’ (very relevant philosophical issue), but on the other hand it is hard for me to see how anyone could dispute that 1 does not equal 0… So to me it seems the issue of `the truth of (any form of) mathematics’ is just not so simple.]]
Andrej Bauer already pointed out that intuitionism is a sound mathematical theory, with enough spin-off to attract also a lot of mathematicians who are not interested in claiming objective philosophical truth for constructivism.
The mathematical reasoning is actually the same, only the principle of the excluded middle (PEM) is not assumed, and infinity is viewed as potential, not as a finished entity [[I disregard some nitty gritty which I believe irrelevant here]]. This does have a profound impact on the theory, although not so similar to the impact of dropping Euclid’s 5th postulate from geometry, since the negation of PEM remains false.
Now to the original question: how do we do mathematics and communicate to each other about it? The only constant that I can see is the amazing phenomenon that we seem largely agreed on the validity of logic as the instrument for mathematical reasoning. And so we are largely agreed as to what is mathematics. But we mathematicians are already unable to reach agreement about what is mathematical infinity, if it exists at all… And then, undoubtedly to me, social and psychological factors come in to decide which view we adopt.
So, to me, large parts of classical mathematics are simply `unrealistic’. PEM does not jibe with my perception of the world in which we live, and strikes me as Dreamland. This for me does not mean that classical mathematics is `wrong’, but it compares for me to chess: an interesting game with little resemblance to real life. [[behold my own psychological factor]]
In contrast, most mathematicians see PEM as an objective truth, and mathematics as idealized science. They are hard put to see any significance in intuitionism/constructivism. This I feel will change however, as the practical advantages of constructivism become clearer.
[[In my recent book Natural Topology (posted earlier on FOM) you can find a simple, applicable (and classically acceptable) framework which contains most of the intuitionistic perspective (I believe).]]
Sorry for the long post, I have only briefly scratched the surface I feel…
—end of message—
Being a moderator is of course not an easy job, and there is something to be said for not allowing this type of discussion. The mission statement says, on the other hand:
contemporary research in foundations of mathematics and which often
do not find their way into journal articles. FOM postings must be
highly relevant to issues and programs in foundations of
mathematics. They should reflect high intellectual and scholarly
standards. However, FOM is not a venue for papers that should be
submitted to journals. Generally, detailed proofs and technical
details are not welcome. Of course, pointers to more extensive
accounts, published in print or on the Web are welcome. Postings
should be thoughtful, well-reasoned, and lively. Although
controversy is both expected and desired, personal invective and
other irrelevant discussions will not be permitted.