We might also reflect that mathematics is not absolute and has a rather more philosophical view of itself.
One of the earliest Greek philosophers refused to countenance any real world use of his philosophies (the educated among you will remind me who). Engineers who so thoroughly use mathematics are using math as a tool and are free from the philosophical considerations.
We might also consider Pythagoras Theorem, even though it is actual a proof for which he is famous. The "theorem" existed long before that, and was a vital engineering tool.
This tells us that while mathematicians are obsessed with proofs that define the limits or extent to which a theorem is true, engineers are concerned with the usefulness in everyday circumstances.
We can reflect that mathematicians are not without the human touch. Fermat’s last Theorem has now been converted into a proof; but no-one seems to comment on Fermat’s claim to have had a proof, perhaps in deference to the esteem in which he is held: the proof has required extensive use of "new" mathematics unavailable to Fermat.
We also should consider that mathematics has its own "king’s new suit of clothes". All proofs are based on previous proofs. “And so ad infinitum?” No, at the heart of mathematics are some axioms. A straight line is the shortest distance between two points etc. Sadly, when it was decided to look at these axioms and see if they too were not capable of being proved, it was found that they could not be. A bit like Schrödinger’s cat? Except that they did open the box and possibly wished they hadn't since they don't know, and now know they can't know, if the cat is alive or dead. This, it seems to me, reduces the whole of mathematics to a great big theoretical structure.
This definition of axioms is a hard one to swallow:
I think it was Russel (
who was involved in this work (still looking).
Euclids work (
suggests that modern math is founded on 2000year old “presumptions”.
So engineers use mathematics as a tool in the practical world. The results are tangible and viable. The mathematics behind them may not be quite so sound? I wonder if we are ever going to see a “Non-euclidian” branch of mathematics, much as we have Newtonian and Non Newtonian.
We already have Null A (non-Aristotelian) thought and this is probably more useful to us as engineers along with Edward De Bono's works on letaral thinking, as a means of "thinking outside of the box".
This, then, bears on the argument that a "greatest" invention or discovery would be something not evolved out of existing thought. That is going to be hard since many "discoveries" or "inventions" appear, with hindsite, almost inevitable once their time has come. Witness the counter claims about Leibnitz and Newton, about who invented TV or the electric light.
Is there an invention that goes beyond conventional mathematics, beyond the current development paths in the accepted diciplines?
Any ideas?
JMW
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