Formal Proofs As Seen From the World Of Physics?
Posted by admin on July 4, 2008
Since high school geometry is typically the first time that a student encounters a formal proof, this can obviously present some difficulties. It can also lead kids to think that two-column proof is the only kind of proof there is – yet that form of proof is almost never used by practicing mathematicians.
It could be easier, if children encountered informal ‘proofs’ in earlier school years, and were required to justify their statements and reasoning. This of course would not be on such formal level as it is in high school, but simply a mindset of teaching mathematics where mathematical statements and truths are justified, there are explanations of where things come from, why something works – and the child also is asked to provide explanations and justifications.
What is your view of formal geometric proofs?
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It has long been commonly assumed that geometric diagrams could only be used as an aid to human intuition and could not be used in rigorous proofs of theorems of Euclidean geometry.
This paper gives a formal system FG whose basic syntactic objects are geometric diagrams and which is strong enough to formalize most if not all of what is contained in the first several books of Euclid’s Elements. This formal system is much more natural than other formalizations of geometry have been.
Most correct informal geometric proofs using diagrams can be translated fairly easily into this system, and formal proofs in this system are not significantly harder to understand than the corresponding informal proofs.
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On the one hand, doing formal proofs forces you to be clear about what depends on what. On the other hand, it forces you to be a bit rigid, because there is an order of priorities: axioms are more basic than theorems, and some theorems depend on other theorems.
There’s nothing really wrong with that, of course, but for most scientific people, it’s not specially useful to continually focus on what is more basic and what is less: In the real world, you may find that you need to switch your assumptions and be flexible. The proofs in Euclidean geometry don’t allow for that, and for that reason I found them somewhat constraining and artificial.
Example: Conservation of energy can first be understood in the context of Newtonian physics as a result of Newton’s laws of motion. But eventually, we have to understand it as a principle in its own right, transcending the confines of Newtonian physics. We no longer think about it in terms of how it was originally introduced.
With regards to education: The problem is always going to be finding the right teachers. From what I have seen, the Japanese do a much better job in encouraging their math teachers to help students understand mathematical reasoning than anyone else.
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