These are the answers by mathematician Toby Bartels to the ten questions about intuitionism:
No.
It's possible to define such a function by making appropriate assumptions, such as the assumption that equality in the real line is decidable. One problem in mathematics is that people don't realise very well that this really is a significant assumption, and that it's optional. Nevertheless, it is a possible and reasonable assumption to make.
Yes.
As normally stated, it doesn't include all of the assumptions. So this is the other side of my response to #1.
No.
Maybe I don't understand what Brouwer was getting at, but I don't see how anybody at all can maintain this! Even for Brouwer, since he would answer Yes to #1, surely the sets R(the real line), R- {0} and R- {0,1} are all different cardinalities (no bijections between them). So there are at least 3 different uncountable cardinalities to him; and I can't think of any other (actually used) form of mathematics that has 3 different cardinalities but not 4 different cardinalities. What am I missing?
Yes.
The last thing that constructivists should be doing is to cede the word "truth value" to the excluded middlers. In intuitionist logic, every meaningful statement has a truth value, not just those statements that are true or false. Since I accept this statement as meaningful, it has a truth value, and furthermore I know what it is: simply "the truth value of the continuum hypothesis".
Yes.
You can make a stronger case that this statement is meaningless. But I accept it as meaningful, so it too has a truth value.
No.
The machine with both lights on is an elegant response. But barring that, this is a futile exercise.
No.
I know what excluded middlers mean when they prove this. But like the Intermediate Value Theorem as normally stated, it's phrased too absolutely to be strictly correct.
Yes.
Of course, sometimes the constructive proof gives no more insight (such as when the obvious classical proof is already constructive!). But sometimes is does give more insight, and it never gives less (assuming both proofs do exist).
Yes.
This is really the most important point of all! Classical reasoning has its place, despite Brouwer's polemics. And constructive reasoning really is useful, despite Hilbert's. Can't we all just get along?
Mu.
Is this supposed to make the same basic point as #9? It doesn't seem to me to be the right way to put it. The formulation of #9 is much better.
Once more, for emphasis:
Mathematics can be done using different kinds of reasoning, and depending on the situation different kinds of reasoning are appropriate.
Amen!