When are two proofs essentially the same? (2007)
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When are two proofs essentially the same? (2007)
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Re: When are two proofs essentially the same? (2007)
#2If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to say it, you are saying the same thing.
Re: When are two proofs essentially the same? (2007)
#3If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
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Re: When are two proofs essentially the same? (2007)
#4> A couple of years ago I spoke at a conference about mathematics that brought together philosophers, psychologists and mathematicians. The proceedings of the conference will appear fairly soon
Can we do better?
Re: When are two proofs essentially the same? (2007)
#5If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
This is like saying that if I walk out of my house, turn right, and walk 10 minutes to the local food store, it's the same as coming out of the house, turning left, and walking 15 minutes around the block. The destination is the same, so surely these are "the same".
I'd argue that this is not the case.
Re: When are two proofs essentially the same? (2007)
#6If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
yes : if two discrete semiotic symbolic networks point to the same signified value they are the same in the way two different poems with the same meaning are the same. which is to say they are unique but have the same meaning.
Re: When are two proofs essentially the same? (2007)
#7If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
A proof is not a statement that something is true, but a demonstration that it is true.
Are you familiar with the proofs-as-programs idea?
The uh, something isomorphism? Idr the name.
Not all programs that implement a function are the same.
When you boil things down to the fundamental steps of the logic you are working on, you needn’t get the same thing.
For one thing, it may be that axioms A and B suffice to prove Z, and that axioms B and C suffice to prove Z, but that B alone doesn’t, and that A and B doesn’t prove C and that B and C doesn’t prove A.
So, the proofs using A and the proofs using C are certainly different.
Re: When are two proofs essentially the same? (2007)
#8If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
This is like saying that if I walk out of my house, turn right, and walk 10 minutes to the local food store, it's the same as coming out of the house, turning left, and walking 15 minutes around the block. The destination is the same, so surely these are "the same". I'd argue that this is not the case.
There's a simple mechanical transformation from one path to the other.
As a proof that "the store is reachable, they are essentially the same if it is already known that you live on a "block" with the store" . If it is not known that you live on a block, then the second proof together with the first gives a much deeper result, proving that you do live on a block. That makes a second proof valuable, but in the monograph of history, it is most parsimonious to make the block proof and the note how it implies to trivially distinct ways of reaching the store.
Re: When are two proofs essentially the same? (2007)
#9> A couple of years ago I spoke at a conference about mathematics that brought together philosophers, psychologists and mathematicians. The proceedings of the conference will appear fairly soon Can we do better?
Not for free we can't.
Re: When are two proofs essentially the same? (2007)
#10If I were allowed a small philosophical leeway, I'd argue that two correct proofs are always the same. For sure they may contain different words or make use of different "abstractions", but it just seems to me that these abstractions should be equivalent if one were willing to unravel it all to a certain degree. Essentially, all proof is, is a statement that says "this is true" and no matter which language you use to…
I disagree with this on two points.
First, oftentimes the interest in proving long-standing, difficult mathematical problems is because we hope a proof will demonstrate new tools for tackling similar problems. In that sense, the exact content of a proof is quite important. Not to mention, there is value in having multiple proofs that each demonstrate quite different toolkits. Mere truth is not often the most important thing -- after all, mathematicians can (and do!) take certain propositions as premises of downstream work. If we discover a proof for one of those premises, that just means we can drop the premise from downstream work. Not having a proof doesn't block us from making use of the proposition anyway.
Second, sometimes the content of the proof is relevant formally. A sibling comment gave an example in terms of paths between two points; it is often the case that you care not only that the points are merely connected, but you also have a preference for which path is taken. Or, you can do an analysis of the paths themselves, and determine their length or average furthest distance from the nearest McDonalds. A path is "just" a proof of connectivity, but the individual paths can be quite distinct when studied themselves. Less abstractly, a constructive proof will yield an algorithm that can be performed, and we know quite well that the variety of sorting algorithms (that "merely" prove that "all lists are sortable") actually vary in quite important ways, including asymptotics and stability.