The application of mathematics is a practical matter; I am not entirely sure what "high philosophy" is, but mathematics is also a conceptual, abstract discipline which can be investigated with no concern whatsoever for practical applications.

The reliance of analysis on Cauchy sequences is not "philosophical gook". zemaj may be naïve in her conclusions (or at least unaware of just how much mathematics one must give up when one takes a constructivist view), but she is hardly alone in her hesitancy about infinity. I wouldn't be sure what to make of the claim that Brouwer, Markov or Martin-Löf didn't know mathematics.

I wouldn't call not understanding the theoretical underpinnings of big-O analysis a "crippling flaw" for a programmer. As long as you understand what it means that an algorithm is O(2^n) and what limitations that entails, then you're grand. If you can also look at an algorithm you've written and work out what its big O complexity is then you know more than most programmers. Both of those can trivially be done without ever touching infinity.

Equally if all you care about is working out if your bridge is going to fall down, you don't really have any need for infinity. In fact you can spend an entire career using math to do all kinds of really awesome things without ever actually understanding math. I'd say that a good 95% of engineers fall into that last category, and I'm certainly not worried about driving over bridges because of it.

That's like eating meat while being morally opposed to killing animals.

How do you figure? If we want to play with food analogies I'd say more like eating meat while not knowing how run a cattle farm.

While disbelieving in cattle farms. That's what the discussion started from.

f ∈ Ο(g(n)) iff ∃c,n₀ ∀n>n₀ : f(n)≤c*g(n)

No infinity required.

As per grandparent comment, the symbol ∀ "can't complete during the life of the universe".

Right. I was about to comment that that's because you're assuming an infinite domain, but the above definition becomes useless unless you do.

The point, though, was that the behavior characterized by the notation is perfectly reasonable even with finite domains (even if the specific notation is not, which was a fatal flaw).