Also, to use infinitesimals rigorously takes a fair amount of knowledge of mathematical logic, otherwise what works and what does not is just magic. Epsilon-delta proofs do not need any magic, nor any more logic than that needed to contend with mathematics at all.
No, to understand why the transfer principle works requires a fair amount of knowledge of mathematical logic. It doesn’t follow that you can’t perform rigorous proofs once you’ve accepted it. Or am I missing something here?
I still disagree. You can use Fermat’s last theorem rigorously without understanding why it works. Same for the four colour theorem. And which mathematics understand why we can classify finite simple groups the way we do? I’d bet fewer than a percent do. Little wonder, if the proof’s 3 volumes long! My point is that there are many theorems a mathematician will use without rigorously knowing why it works. Oh sure, you can tell them a rough story outlining the ideas. But could the prove it themselves? Probably not, without a deep understanding of the area. Yet even without that understanding, they can use these theorems in formal proofs. They can get a machine to check over it.
Now, I admit that’s unsatisfying. I agree that if they don’t, then they don’t have a rigorous understanding of the theorem. Eventually, problems will arise which they cannot resolve without understanding that which they accepted as magic. But is that really so fatal a flaw for teaching students the hyperreals? One only needs a modest amount of logic, perhaps enough for a course or two, to understand why the transfer principle works. Which seems a pretty good investment, given how much model theory sheds light on what we take for grounded.
Now I suppose if you find infinitary mathematics ugly, then is all besides the point. And unfortunately, there’s not much I can say against that beyond the usual arguements and personal aesthetics.
You can understand what these theorems say without knowing how they were proved. But non-standard analysis requires a substantial amount of extra knowledge to even understand the transfer principle. In contrast, epsilon-delta requires no such sophistication.
To stay on computer science analogies, this reminds me of the principle of abstraction. When you call an API, it sort of feels like magic. A task gets done, and you trust that it was done correctly, and that saves you the time of controlling the code and rewriting it from scratch. “We have only to think out how this is to be done once, and forget then how it is done.” (A. Turing, 1947).
Also, to use infinitesimals rigorously takes a fair amount of knowledge of mathematical logic, otherwise what works and what does not is just magic. Epsilon-delta proofs do not need any magic, nor any more logic than that needed to contend with mathematics at all.
No, to understand why the transfer principle works requires a fair amount of knowledge of mathematical logic. It doesn’t follow that you can’t perform rigorous proofs once you’ve accepted it. Or am I missing something here?
If you don’t understand why the transfer principle works, you would just be accepting it as magic. This is not rigorous.
I still disagree. You can use Fermat’s last theorem rigorously without understanding why it works. Same for the four colour theorem. And which mathematics understand why we can classify finite simple groups the way we do? I’d bet fewer than a percent do. Little wonder, if the proof’s 3 volumes long! My point is that there are many theorems a mathematician will use without rigorously knowing why it works. Oh sure, you can tell them a rough story outlining the ideas. But could the prove it themselves? Probably not, without a deep understanding of the area. Yet even without that understanding, they can use these theorems in formal proofs. They can get a machine to check over it.
Now, I admit that’s unsatisfying. I agree that if they don’t, then they don’t have a rigorous understanding of the theorem. Eventually, problems will arise which they cannot resolve without understanding that which they accepted as magic. But is that really so fatal a flaw for teaching students the hyperreals? One only needs a modest amount of logic, perhaps enough for a course or two, to understand why the transfer principle works. Which seems a pretty good investment, given how much model theory sheds light on what we take for grounded.
Now I suppose if you find infinitary mathematics ugly, then is all besides the point. And unfortunately, there’s not much I can say against that beyond the usual arguements and personal aesthetics.
You can understand what these theorems say without knowing how they were proved. But non-standard analysis requires a substantial amount of extra knowledge to even understand the transfer principle. In contrast, epsilon-delta requires no such sophistication.
To stay on computer science analogies, this reminds me of the principle of abstraction. When you call an API, it sort of feels like magic. A task gets done, and you trust that it was done correctly, and that saves you the time of controlling the code and rewriting it from scratch. “We have only to think out how this is to be done once, and forget then how it is done.” (A. Turing, 1947).