I am an engineer and know nothing about philosophy, my friend is an economist who argues that the problem with induction invalidates all predictions based on induction.
I can understand this on a fundamental level but does this invalidate all predictions and models made from science? He argues that the scientific method is not truly deductive and so invalid. It seems to me that we have gotten pretty far with these predictions so does it matter if they are invalid?
Science does not work directly from induction, it requires a theory in addition to observations. That provides a mechanism that forces statistics to apply. As you repeat an experiment, the odds it will ultimately fail to repeat in the future do go down. They never get to zero. You could still be wrong. But that becomes less and less likely as you successfully reuse the theory over time.
Even if you don't do the p-value computations, you are safely covered by the Rule of Succession, as long as you have a falsifiable theory, and you challenge it. You have some odds p < 1 of being wrong, and if you successfully repeatedly use your result n times the odds it will eventually fail are <= p^n, which converges to zero when n is infinite. Since 'being falsifiable' implies you would recognize when you failed, you are legitimately sampling a distribution, and this is not induction, it is math.
Each time you 'fail to falsify' you 'reject the null hypothesis' and in doing so you are building a statistical basis for believing your model and its underlying assumptions are less and less likely to be wrong to a significant degree. If you do keep actually compute the p-values, as people in very slippery sciences like psychology and sociology force themselves to do, you can go back and figure out just how unlikely ultimate failure is becoming due to seeing the outcome in various instantiations.
Statistical convergence is not really induction in the pure philosophical sense, it is deduction with probabilistic truth values, that never really reaches 100% reliability.
So he is 'right' that science is never logically watertight. And we already knew that, given that it constantly changes and evolves. But he is wrong to say the procedure is not deductive.
...does [the problem of induction] invalidate all predictions and models made from science?
Yes. The problem says that “we are not entitled to any degree of confidence whatever, no matter how slight, in any predictions regarding what we have not observed” Lange (2011), "Hume and the Problem of Induction", p. 43.
The sticking point is the problem's underlying premise: there is no reason to believe that similar events will have similar results; instead, from any event, any result is conceivable. That premise, of course, destroys predictability.
...so does it matter if [the predictions] are invalid?
No. Induction is famously the "glory of Science" and the "scandal of Philosophy" (C.D. Broad 1926). Science has assumed, successfully, that similar events will indeed have similar results. The predictions are not invalid in the sense that they are wrong; it is just that the philosophical justification is not present. As a practical matter, the predictions are reliable.
The Problem of Induction does represent a problem for the philosophy that is often referred to as science. That empirical reasoning is not "proven" may or may not be a problem for you personally. Empiricism has lead to knowledge that is closer to true then what we had before. So empiricism is useful as the computer you are using to view this demonstrates.
Is our current understanding of empiricism "true"? Almost certainly not. That is part of the philosophy of the sciences. The assumption that our current understanding is flawed in some way. This encourages people to suggest improvements. As a result our understanding improves over time.
So no I do not have an issue with the problem of induction, it only represent another area where our understanding could improve. That science currently has flaws in its understanding of things is a strength, rather than a weakness.
