Another reason is that in engineering, people don't get situations that perfectly match the axioms of any theorem, as we never have rulers or gadgets that measure any length or weight to an infinite number of decimal places. We're always rounding off in real life.
But the logic of mathematics deals in abstract absolutes. So you can never really get any real-life situation in which any mathematical theorem can ever apply. So that means that all scientific theories built on maths have to be wrong, right?
Well, yes. BUT...if you understand the proof, you can add an error margin to your premises, and follow the theorem's logic through to its conclusion, with the error margins. Thus, you can calculate the error margin in the conclusion, from the error margin in your initial premises. Using that, you can then decide what error margin you need in your conclusions, and then calculate the error margin in your initial premises, i.e. you can then calculate how accurate you need to measure your premises, to get that level of accuracy in your conclusion.
OK. That makes sense. But then why not include that in your theorem? Well, you can. But then you've fixed the error margins.
What if you have multiple premises, and with one premise you can be extremely accurate, and in another you can't be all that accurate? Well, if you understand the theorem's logic, you can still work out a way to be as accurate as you need in your conclusion.
But if you don't understand the theorem's logic, then you can only stick to what you were taught, and then you can only use error margins according to the way you were taught, and then the above situation where you can't be that accurate in one measurement, but you can be very accurate in another, is a situation in which you simply cannot use the theorem. If you do use the theorem in such a situation, then you'll get the wrong answer. Then if you make a vaccine with that situation, your vaccine may be a lot less accurate than you realise.
Why? BECAUSE ERRORS MULTIPLY. They often don't add together. They MULTIPLY.
Unless you understand the logic behind a theorem, it is incredibly easy to get a result that will kill millions of people in a single year, when a slightly different approach would have resulted in only the deaths of a few hundred.
This is a problem that occurs a LOT nowadays. Someone develops a theorem or some logic that proves a conclusion under certain conditions. A politician hears about it, and thinks that it would be useful in the current situation. So he goes ahead, promises to solve major problems and gets elected because of it. Then it is applied. Everyone cheers. Then a few years later, reports highlight strange results. Then 10 years later, a calamity happens. No-one knows why.
The reason is often that the original logic made sense, in certain conditions. But the precise conditions in which the politician's idea was implemented, was extremely similar to the conditions of the theorem, but not quite, and the difference between them just so happened to mean the theorem's logic would not follow exactly in the real-life situation, and so the conclusion wouldn't be guaranteed to happen, and anything could have happened, and in this case, a calamity.
Now, had the politician known this in advance, of course he would not have done it. But the premises of his situation were so eerily close to the axioms, and only had to be tweaked ever so slightly because of practicalities and political considerations, that no-one thought it would ever make any difference.
Had the politician and his advisors learned the proof, and understood it, they would have realised this immediately. But they didn't, because soundness is considered to be a different issue to validity. So they thought that their argument was sound, when it wasn't, because they didn't properly understand the validity of the logic and how it actually worked.
An apt analogy of the difference between validity and soundness as validity = "comprehension of an argument" and soundness as "memorisation of an argument". Soundness means you know the formula. But you don't really understand why it's true.
Now wait. This all makes me sound very clever, doesn't it, as if I understand all this much better than anyone else, right?
Actually, the only reason I know this, is because my maths teacher told us this and made us prove it. He explained that in maths exams for our qualifications for graduating high school, have long questions. So you'd have a part, and then use the answer for the 1st part, in the 2nd part, and so on. Most teachers would let students use a calculator for each interim calculation. But he pointed out that calculators would only calculate to so many digits, and that ERRORS MULTIPLY, and so if we used calculators to calculate the interim results, our end results could be way off, and then we would lose marks for getting the wrong answer.
So he MADE US DO ALL OUR MATHS QUESTIONS WITHOUT USING A CALCULATOR! That made all our homeworks and mock exams much harder than all the other students. But you know what? He was right. We all got As for every exam he prepared us for, bar one person who didn't do all the homeworks, and even he got an A and a C.
I sometimes used to test his claim by using a calculator in the practise exams he gave us, just to see what would happen. A lot of the time, it didn't make much of a difference. But in some exam questions, I would have been way off if I had used a calculator.
So I learned from that.
Unforunately, not everyone had a maths teacher who was like that.
