The two are related in two ways. First the mathematical case for zero tax on capital income states that as time goes to infinity the tax rate goes to zero. Many economists pretend this means that it should be cut to zero right now. This is utter nonsense.
The argument is that we wish people to have high consumption. For that to be possible they must save and invest. Capital income taxation penalizes saving and so prevents capital accumulaion -- by causing higher consumption now. The whole argument is based on the fact that, when one has a budget constraint, more now means less later. But this fact that the whole discussion is about doing the opposite now of what you want to do in the future is conveniently forgotten when it shifts from assumptions about desired consumption to conclusions about optimal taxation. I do not think that this can be an honest mistake.
The solution to the simplest version of the problem is that a state which wished to take wealth from investors and worries about their incentive to save should take the wealth as quickly as it can until it has taken all that it has any desire to take, then stop. The tax on capital income goes to zero if and only if the state does not wish to take from capitalists. Exactly when a lump sum tax which capitalists must pay no matter what would also be zero. As claimed, there is no tradeoff in the very long run between concern about incentives and other aims. But the reason is that the other aims are totally achieved as they would be if there were no problem with incentives.
The math is fairly simple (pdf warning). A slightly different and more realistic model has a more extreme result. The state takes from capitalists at time t even though it would rather let them keep the wealth in order to reduce their consumption at earlier times.
Here the trick is to assume the future is now.
A much more serious problem which actually affects political debate is the assumption that tax cuts will not lead to an exploding debt. This is just and assumption and it is part of the rhetoric of advocates of tax cuts, but it is also true of the models used by serious economists to analyse tax cuts. The reason is that the standard approach requires the economy to converge to a steady state. An exploding debt to gdp ratio is not allowed, because all such ratios are must converge to constants in all respectable models. Here the point is that this is required for the model to be respectable -- it does not follow from other core assumptions.
So, for technical reasons, it is assumed that the tax cut is paid for either by a tax increase or a spending cuts (often the assumed tax increase is a lump sum tax) which does not affect incentives. This makes the analysis useless. A more realistic assumption would be that the exploding growth will cause a future policy change involving taxes which have actually been collected without causing uprisings. This means that low taxes now imply high taxes in the future. If one considers only capital income taxes, it means the believe that they should go to zero as time goes to infinity implies they should be high now. If the choice is taxes in the future or in the present (and it is) then arguments that taxes should be low in the future imply they should be high in the present.
Of course the enthusiasts for tax cuts hope that the debt will lead to cuts in social welfare spending. But they don't admit it, because even discussing the possibility of such cuts in the future is political poison. Also, I think, they have short planning horizons and want low taxes at the time of the next election and don't care about the long run (but are willing to use analysis of the long run whenever it is convenient).v The assumption that something will be done to deal with the debt (and that this something won't have bad incentive effects) is almost always combined with the insane assumption of Ricardian equivalence that ordinary people keep track of the national debt, know what share will be paid by them and their heirs and adjust consumption accordingly. In the real world, government bonds create the illusion of wealth causzing higher consumption and crowding out investment.v I think there is another problem with journalistic presentation of the debate. Supply side loons say tax cuts will cause so much growth that they pay for themselves. Sensible people say this is nonsense. Ballanced journalists say the truth is probably somewhere in between -- they will cause more growth but not enough to pay for themselves. This means they can say that, on the one hand GDP will be higher and on the other the national debt will be higher. They don't explain why the second is a bad thing (people just assume it is). But if it is a bad thing, it is because the debt causes lower GDP (it isn't painful to carry it in itself). There is a logical contradiction between believing public debt is bad because it is bad for GDP growth and ignoring this effect when discussing GDP growth. This is so obvious that I think the popularity of pairs of inconsistent claims must be do to something very strong. I think it is the need to find something good to say about Republicans, which regularly drives US journalists crazy.
So the discussion suffers from two huge fatal errors based on playing around with the long run.
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