I am wondering why in "There's no point in doing that" we use the preposition in but in "There's no reason to do that" we use the preposition to?
The short answer is that in this case to is not a preposition—it's an infinitive marker, which tells you that the immediately following verb is to be construed as an infinitive.
However, with the gerund (doing in your example) or any other nominal you may use either preposition: There's no point in flying = There's no point to flying. in is far more usual, but to is unremarkable.
EDIT OP asks how we may know that to is an infinitive marker in no reason to do that.
First, any preposition requires a nominal—a noun, a pronoun, a gerund, or a noun phrase—as its object.
Consider then the following sentences. (In order to keep the constructions parallel I start with instances of prepositional to with reason, a usage which is uncommon today, but acceptable.)
There's no reason to that. ... that is a pronoun, the object of the preposition to, so we can separate it from to and move it to the head of a sentence and say That is irrational.
There's no reason to doing that. ... doing is a nominal, a gerund, and doing that is a nominal phrase—again, it's the object of the preposition to, so we can say Doing that is irrational.
There's no reason to do that. ... The bare infinitive form do, and the phrase do that, cannot be employed nominally; we cannot parse do that as the object of the preposition to and say *Do that is irrational, which is meaningless.
We must parse to as an infinitive marker on do and employ the two together to produce the acceptable sentence To do that is irrational.
This question conflates two separate distinctions, that need to be addressed separately to gain any clarity:
- Point (purpose or teleology) vs Reason (motivation or justification)
- in doing (the gerund form of the verb) vs to do (the infinitive form)
A. There's no
point in doing this --- There's no
reason for doing this.
The nouns point and reason produce the different prepositions in and for. I'd submit that, on the one hand, the point (end purpose) is presumed to be found contained within the doing of something, in the sense that it (the end purpose) should emerge as a result of the act of doing. On the other hand, the reason (motivation) is something that comes from you, not from the doing itself, but which you hold out for the act of doing.
B. There's no reason
in doing this --- There's no reason
to do this
As StoneyB's answer points out, the forms of the verb here imply a subtle semantic difference. (Note that I'm deliberately using reason in both sentences here, in order to isolate the difference produced by the verb forms.) On the one hand, the gerund form doing of the verb, combined with the preposition in, implies a present tense, as if the doing were underway. The word reason in this context takes on the meaning of sense, as in, there is no sense, no sound thought contained in the doing of this.
On the other hand, the infinitive to do connotes a future tense (it denotes atemporal, but we read it as a doing yet to come); and reason in this context again takes on the meaning of motivation or justification. There's no justification for undertaking the doing of this.
This brings me back to for doing, which, in combining the gerund form of the verb doing with the preposition for (which, remember, I also identified as connoting justification), implies a judgement for a doing that is already underway (or on the precipice): there's no reason for doing this is a disapproving judgement after-the-fact.