You’ve probably heard of circular reasoning. You probably are familiar with the dilemma of the chicken and the egg, “which came first?”
Before beginning, the purpose here is to create a bit of amusement and to get the reader to think about math proofs.
The issue here is the calculation of the integral of a monomial to get the area.
You have a choice: do you want to use a known area to confirm a calculation, or do you want to use a confirm calculation to prove that a reported value for the area is correct?
- C- we can test and confirm the calculation
- A- the area value is known
Herein lies the dilemma:
As a math student you have type of experience using both C–>A and C to get A. The textbook gives you a confirmed calculation so we can say that C is true… (warning background music begins to play)
Then one day, skeptics find some errors in the calculus textbook and a judge rules that the calculus textbook is fallible, and we can’t use it to validate calculus calculations (see Appendix B). We must prove a calculation in some way before we can use it.
Suddenly, someone offers you a lot of money–$$$$–if you can prove that the calculation for the derivative of a monomial is correct. Money? Okay I now have your complete attention.
If we have the area, that makes A true and that, with A–>C makes C true.
Let me give a quick example of this, maybe we do something like we have a printer that prints out the curve shape and print out a square we know the area of the square so we can correlate paperweight to pay for area and from that we can get an estimate of what the area is. Now if we show that the calculation gives an answer which is really really close to what we get from doing our paper weigh experiment–and this can be done using an electronic balance that goes to five digits–we might convince the person offering the money that we have done enough for their purposes, and thus they should pay us.
Appendix A
I’m getting an uneasy feeling about parts of this. Propositional logic may not fit with what I’m trying to do.
I went back and rewrote A and C. I did this because…
A –> C
The above “can” work, but it isn’t a guaranteed. It’s entirely possible that we made a mistake and we don’t have the correct calculation, and when we take our known area and test it with our calculation, it doesn’t work.
I think in propositional logic you write an implication (p –> q) only if EVERY time the p is true then the q is true.
Appendix B
The story is unfair in a way because textbooks show a proof for an idea along with introduced in it and tell me what the calculation is. Those studebt with more advancement mathematics can follow the proof and this makes it easier to accept the calculation
Rob Sterling 