Dear Israel Perez,
I enjoyed reading your essay.
Your discussion of negative solutions was very thought provoking. I think it is worth pointing out that sometimes negative solutions should not be dismissed and
do have significant physical meaning. Dirac's finding of the positron comes to mind.
I wonder if all fake negative solutions would go away if we dealt with the right mathematics. For example, we always thing of the whole set of real numbers. Maybe we should do mathematics only with positive real numbers. We always deal with groups. Maybe we should deal with the less familiar monoids.
I also enjoyed you stressing the importance of understanding. I once humorously pointed out to my thesis advisor, Alex Heller, that
there are subatomic particles in nature that follow equations of motion
that human beings cannot solve. And even though humans do not know
where the particles will go, the particles seem to know exactly where to
go. Professor Heller responded by saying that this shows that science has
nothing to do with calculating or predicting. Calculations can be done by
computers. Predictions can be performed by subatomic particles. Science
is about understanding -- an ability only human beings possess.
Again, thanks for a great essay.
All the best,
Noson Yanofsky